The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS...
Transcript of The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS...
![Page 1: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/1.jpg)
The Cross Product
In this section, we will learn about:
Cross products of vectors
and their applications.
![Page 2: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/2.jpg)
THE CROSS PRODUCT
The cross product a x b of two vectors a and b, unlike the dot product, is a vector.
§ For this reason, it is also called the vector product.
§ Note that a x b is defined only when a and b are three-dimensional (3-D) vectors.
![Page 3: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/3.jpg)
If a = ‹a1, a2, a3› and b = ‹b1, b2, b3›, then the cross product of a and b is the vector
a x b = ‹a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1›
THE CROSS PRODUCT Definition 1
![Page 4: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/4.jpg)
This may seem like a strange way of defining a product.
CROSS PRODUCT
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The reason for the particular form of Definition 1 is that the cross product defined in this way has many useful properties, as we will soon see.
§ In particular, we will show that the vector a x bis perpendicular to both a and b.
CROSS PRODUCT
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In order to make Definition 1 easier to remember, we use the notation of determinants.
CROSS PRODUCT
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A determinant of order 2 is defined by:
§ For example,
a bad bc
c d= -
2 12(4) 1( 6) 14
6 4= - - =
-
DETERMINANT OF ORDER 2
![Page 8: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/8.jpg)
A determinant of order 3 can be defined in terms of second-order determinants as follows:
1 2 32 3 1 3 1 2
1 2 3 1 2 32 3 1 3 1 2
1 2 3
a a ab b b b b b
b b b a a ac c c c c c
c c c= - +
DETERMINANT OF ORDER 3 Equation 2
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Observe that:§ Each term on the right side of Equation 2 involves
a number ai in the first row of the determinant.§ This is multiplied by the second-order determinant
obtained from the left side by deleting the row and column in which it appears.
DETERMINANT OF ORDER 3
1 2 32 3 1 3 1 2
1 2 3 1 2 32 3 1 3 1 2
1 2 3
a a ab b b b b b
b b b a a ac c c c c c
c c c= - +
![Page 10: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/10.jpg)
Notice also the minus sign in the second term.
DETERMINANT OF ORDER 3
1 2 32 3 1 3 1 2
1 2 3 1 2 32 3 1 3 1 2
1 2 3
a a ab b b b b b
b b b a a ac c c c c c
c c c= - +
![Page 11: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/11.jpg)
For example,
1 2 10 1 3 1 3 0
3 0 1 1 2 ( 1)4 2 5 2 5 4
5 4 21(0 4) 2(6 5) ( 1)(12 0)
38
-= - + -
- --
= - - + + - -= -
DETERMINANT OF ORDER 3
![Page 12: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/12.jpg)
Now, let’s rewrite Definition 1 using second-order determinants and the standard basis vectors i, j, and k.
CROSS PRODUCT
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We see that the cross product of the vectors a = a1i +a2j + a3k and b = b1i + b2j + b3kis:
2 3 1 3 1 2
2 3 1 3 1 2
a a a a a ab b b b b b
´ = - +a b i j k
CROSS PRODUCT Equation 3
![Page 14: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/14.jpg)
In view of the similarity between Equations 2 and 3, we often write:
1 2 3
1 2 3
a a ab b b
´ =i j k
a b
CROSS PRODUCT Equation 4
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The first row of the symbolic determinant in Equation 4 consists of vectors.
§ However, if we expand it as if it were an ordinary determinant using the rule in Equation 2, we obtain Equation 3.
CROSS PRODUCT
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The symbolic formula in Equation 4 is probably the easiest way of remembering and computing cross products.
CROSS PRODUCT
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If a = <1, 3, 4> and b = <2, 7, –5>, then
1 3 42 7 5
3 4 1 4 1 37 5 2 5 2 7( 15 28) ( 5 8) (7 6)
43 13
´ =-
= - +- -
= - - - - - + -= - + +
i j ka b
i j k
i j ki j k
CROSS PRODUCT Example 1
![Page 18: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/18.jpg)
Show that a x a = 0 for any vector a in V3.
§ If a = <a1, a2, a3>, then
CROSS PRODUCT Example 2
1 2 3
1 2 3
2 3 3 2 1 3 3 1
1 2 2 1
( ) ( )( )
0 0 0
a a aa a aa a a a a a a a
a a a a
´ =
= - - -
+ -= - + =
i j ka a
i jk
i j k 0
![Page 19: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/19.jpg)
One of the most important properties of the cross product is given by the following theorem.
CROSS PRODUCT
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The vector a x b is orthogonal to both a and b.
CROSS PRODUCT Theorem
![Page 21: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/21.jpg)
In order to show that a x b is orthogonal to a, we compute their dot product as follows
CROSS PRODUCT Proof
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2 3 1 3 1 21 2 3
2 3 1 3 1 2
1 2 3 3 2 2 1 3 3 1 3 1 2 2 1
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
( )
( ) ( ) ( )
0
a a a a a aa a a
b b b b b ba a b a b a a b a b a a b a ba a b a b a a a b b a a a b a b a a
´ ×
= - +
= - - - + -
= - - + + -
=
a b aCROSS PRODUCT Proof
![Page 23: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/23.jpg)
A similar computation shows that (a x b) · b = 0
§ Therefore, a x b is orthogonal to both a and b.
CROSS PRODUCT Proof
![Page 24: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/24.jpg)
Let a and b be represented by directed line segments with the same initial point, as shown.
CROSS PRODUCT
![Page 25: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/25.jpg)
Then, Theorem 5 states that the cross product a x b points in a direction perpendicular to the plane through a and b.
CROSS PRODUCT
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It turns out that the direction of a x b
is given by the right-hand rule, as follows.
CROSS PRODUCT
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If the fingers of your right hand curl in the direction of a rotation (through an angle less than 180°) from a to b, then your thumb points in the direction of a x b.
RIGHT-HAND RULE
![Page 28: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/28.jpg)
We know the direction of the vector a x b.
The remaining thing we need to complete its geometric description is its length |a x b|.
§ This is given by the following theorem.
CROSS PRODUCT
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If θ is the angle between a and b(so 0 ≤ θ ≤ π), then
|a x b| = |a||b| sin θ
CROSS PRODUCT Theorem 6
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From the definitions of the cross product and length of a vector, we have:
|a x b|2
= (a2b3 – a3b2)2 + (a3b1 – a1b3)2 + (a1b2 – a2b1)2
= a22b3
2 – 2a2a3b2b3 + a32b2
2 + a32b1
2
– 2a1a3b1b3 + a12b3
2 + a12b2
2
– 2a1a2b1b2 + a22b1
2
CROSS PRODUCT Proof
![Page 31: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/31.jpg)
= (a12 + a2
2 + a32)(b1
2 + b22 + b3
2) – (a1b1 + a2b2 + a3b3)2
= |a|2|b|2 – (a . b)2
= |a|2|b|2 – |a|2|b|2 cos2θ [Th. 3 in Sec. 12.3]
= |a|2|b|2 (1 – cos2θ)
= |a|2|b|2 sin2θ
CROSS PRODUCT Proof
![Page 32: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/32.jpg)
Taking square roots and observing that because sin θ ≥ 0 when
0 ≤ θ ≤ π, we have:
|a x b| = |a||b| sin θ
2sin sinq q=
CROSS PRODUCT Proof
![Page 33: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/33.jpg)
A vector is completely determined by its magnitude and direction.
Thus, we can now say that a x b is the vector that is perpendicular to both a and b, whose:
§ Orientation is determined by the right-hand rule
§ Length is |a||b| sin θ
CROSS PRODUCT
![Page 34: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/34.jpg)
In fact, that is exactly howphysicists define a x b.
CROSS PRODUCT
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Two nonzero vectors a and b are parallel if and only if
a x b = 0
CROSS PRODUCT Corollary 7
![Page 36: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/36.jpg)
Two nonzero vectors a and b are parallel if and only if θ = 0 or π.
§ In either case, sin θ = 0.
§ So, |a x b| = 0 and, therefore, a x b = 0.
CROSS PRODUCT Proof
![Page 37: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/37.jpg)
The geometric interpretation of Theorem 6 can be seen from this figure.
CROSS PRODUCT
![Page 38: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/38.jpg)
If a and b are represented by directed line segments with the same initial point, then they determine a parallelogram with base |a|, altitude |b| sin θ, and area A = |a|(|b| sin θ)
= |a x b|
CROSS PRODUCT
![Page 39: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/39.jpg)
Thus, we have the following way of interpreting the magnitude of a cross product.
CROSS PRODUCT
![Page 40: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/40.jpg)
The length of the cross product a x bis equal to the area of the parallelogram determined by a and b.
CROSS PRODUCT MAGNITUDE
![Page 41: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/41.jpg)
Find a vector perpendicular to the plane that passes through the points
P(1, 4, 6), Q(-2, 5, -1), R(1, -1, 1)
CROSS PRODUCT Example 3
![Page 42: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/42.jpg)
The vector is perpendicular to both and .
§ Therefore, it is perpendicular to the plane through P, Q, and R.
PQ PR´uuur uuur
PQuuur
PRuuur
CROSS PRODUCT Example 3
![Page 43: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/43.jpg)
From Equation 1 in Section 12.2, we know that:
( 2 1) (5 4) ( 1 6)3 7
(1 1) ( 1 4) (1 6)5 5
PQ
PR
= - - + - + - -= - + -
= - + - - + -= - -
i j ki j k
i j kj k
uuur
uuur
CROSS PRODUCT Example 3
![Page 44: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/44.jpg)
We compute the cross product of these vectors:
3 1 70 5 5
( 5 35) (15 0) (15 0)40 15 15
PQ PR´ = - -- -
= - - - - + -= - - +
i j k
i j ki j k
uuur uuur
CROSS PRODUCT Example 3
![Page 45: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/45.jpg)
Therefore, the vector ‹-40, -15, 15› is perpendicular to the given plane.
§ Any nonzero scalar multiple of this vector, such as ‹-8, -3, 3›, is also perpendicular to the plane.
CROSS PRODUCT Example 3
![Page 46: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/46.jpg)
Find the area of the triangle with vertices
P(1, 4, 6), Q(-2, 5, -1), R(1, -1, 1)
CROSS PRODUCT Example 4
![Page 47: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/47.jpg)
In Example 3, we computed that
§ The area of the parallelogram with adjacent sides PQ and PR is the length of this cross product:
2 2 2( 40) ( 15) 15 5 82PQ PR´ = - + - + =uuur uuur
CROSS PRODUCT Example 4
40, 15,15PQ PR´ = - -uuur uuur
![Page 48: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/48.jpg)
The area A of the triangle PQRis half the area of this parallelogram, that is:
52 82
CROSS PRODUCT Example 4
![Page 49: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/49.jpg)
If we apply Theorems 5 and 6 to the standard basis vectors i, j, and k using θ = π/2, we obtain:
i x j = k j x k = i k x i = j
j x i = -k k x j = -i i x k = -j
CROSS PRODUCT
![Page 50: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/50.jpg)
Observe that: i x j ≠ j x i
§ Thus, the cross product is not commutative.
CROSS PRODUCT
![Page 51: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/51.jpg)
Also, i x (i x j) = i x k = -j
However, (i x i) x j = 0 x j = 0
§ So, the associative law for multiplication does not usually hold.
§ That is, in general, (a x b) x c ≠ a x (b x c)
CROSS PRODUCT
![Page 52: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/52.jpg)
However, some of the usual laws of algebra do hold for cross products.
CROSS PRODUCT
![Page 53: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/53.jpg)
The following theorem summarizes the properties of vector products.
CROSS PRODUCT
![Page 54: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/54.jpg)
If a, b, and c are vectors and c is a scalar, then
1. a x b = –b x a
2. (ca) x b = c(a x b) = a x (cb)
3. a x (b + c) = a x b + a x c
CROSS PRODUCT PROPERTIES Theorem 8
![Page 55: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/55.jpg)
4. (a + b) x c = a x c + b x c
5. a · (b x c) = (a x b) · c
6. a x (b x c) = (a · c)b – (a · b)c
CROSS PRODUCT PROPERTIES Theorem 8
![Page 56: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/56.jpg)
These properties can be proved by writing the vectors in terms of their components and using the definition of a cross product.
§ We give the proof of Property 5 and leave the remaining proofs as exercises.
CROSS PRODUCT PROPERTIES
![Page 57: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/57.jpg)
Let a = <a1, a2, a3>
b = <b1, b2, b3>
c = <c1, c2, c3>
CROSS PRODUCT PROPERTY 5 Proof
![Page 58: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/58.jpg)
Then, a · (b x c) = a1(b2c3 – b3c2) + a2(b3c1 – b1c3)
+ a3(b1c2 – b2c1)= a1b2c3 – a1b3c2 + a2b3c1 – a2b1c3
+ a3b1c2 – a3b2c1
= (a2b3 – a3b2)c1 + (a3b1 – a1b3)c2
+ (a1b2 – a2b1)c3
=(a x b) · c
CROSS PRODUCT PROPERTY 5 Proof—Equation 9
![Page 59: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/59.jpg)
SCALAR TRIPLE PRODUCT
The product a . (b x c) that occurs in Property 5 is called the scalar triple product of the vectors a, b, and c.
![Page 60: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/60.jpg)
Notice from Equation 9 that we can write the scalar triple product as a determinant:
1 2 3
1 2 3
1 2 3
( )a a ab b bc c c
× ´ =a b c
SCALAR TRIPLE PRODUCTS Equation 10
![Page 61: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/61.jpg)
The geometric significance of the scalar triple product can be seen by considering the parallelepiped determined by the vectors a, b, and c.
SCALAR TRIPLE PRODUCTS
![Page 62: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/62.jpg)
The area of the base parallelogram is:
A = |b x c|
SCALAR TRIPLE PRODUCTS
![Page 63: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/63.jpg)
If θ is the angle between a and b x c, then the height h of the parallelepiped is:
h = |a||cos θ|
§ We must use |cos θ| instead of cos θ in caseθ > π/2.
SCALAR TRIPLE PRODUCTS
![Page 64: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/64.jpg)
Hence, the volume of the parallelepiped is:
V = Ah= |b x c||a||cos θ| = |a · (b x c)|
§ Thus, we have proved the following formula.
SCALAR TRIPLE PRODUCTS
![Page 65: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/65.jpg)
The volume of the parallelepiped determined by the vectors a, b, and c is the magnitude of their scalar triple product:
V = |a ·(b x c)|
SCALAR TRIPLE PRODUCTS Formula 11
![Page 66: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/66.jpg)
If we use Formula 11 and discover that the volume of the parallelepiped determined by a, b, and c is 0, then the vectors must lie in the same plane.
§ That is, they are coplanar.
COPLANAR VECTORS
![Page 67: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/67.jpg)
Use the scalar triple product to show that the vectors
a = <1, 4, -7>, b = <2, -1, 4>, c = <0, -9, 18>
are coplanar.
COPLANAR VECTORS Example 5
![Page 68: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/68.jpg)
We use Equation 10 to compute their scalar triple product:
1 4 7( ) 2 1 4
0 9 18
1 4 2 4 2 11 4 7
9 18 0 18 0 91(18) 4(36) 7( 18) 0
-× ´ = -
-
- -= - -
- -
= - - - =
a b c
COPLANAR VECTORS Example 5
![Page 69: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/69.jpg)
Hence, by Formula 11, the volume of the parallelepiped determined by a, b, and c is 0.
§ This means that a, b, and c are coplanar.
COPLANAR VECTORS Example 5
![Page 70: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/70.jpg)
The product a x (b x c) that occurs in Property 6 is called the vector triple product of a, b, and c.
VECTOR TRIPLE PRODUCT
![Page 71: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/71.jpg)
CROSS PRODUCT IN PHYSICS
The idea of a cross product occurs often in physics.
![Page 72: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/72.jpg)
CROSS PRODUCT IN PHYSICS
In particular, we consider a force Facting on a rigid body at a point given by a position vector r.
![Page 73: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/73.jpg)
For instance, if we tighten a bolt by applying a force to a wrench, we produce a turning effect.
CROSS PRODUCT IN PHYSICS
![Page 74: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/74.jpg)
The torque τ (relative to the origin) is defined to be the cross product of the position and force vectors
τ = r x F
§ It measures the tendency of the body to rotate about the origin.
TORQUE (Moment of Force)
![Page 75: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/75.jpg)
The direction of the torque vector indicates the axis of rotation.
TORQUE
![Page 76: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/76.jpg)
According to Theorem 6, the magnitude of the torque vector is
|τ | = |r x F| = |r||F| sin θ
where θ is the angle between the position and force vectors.
TORQUE
![Page 77: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/77.jpg)
Observe that the only component of Fthat can cause a rotation is the one perpendicular to r—that is, |F| sin θ.
§ The magnitude of the torque is equal to the area of the parallelogram determined by r and F.
TORQUE
![Page 78: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/78.jpg)
Compare to this.Torque: BioMechanics
VS.
![Page 79: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/79.jpg)
A bolt is tightened by applying a 40-N force to a 0.25-m wrench, as shown.
§ Find the magnitude of the torque about the center of the bolt.
§ Find the torque vector with n is a unit vector heading down into the slide.
§ Solve by hand and using Matlab. Compare
TORQUE Homework
![Page 80: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/80.jpg)
Homework Solution
![Page 81: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/81.jpg)
The magnitude of the torque vector is:
|τ| = |r x F| = |r||F| sin 75°= (0.25)(40) sin75°= 10 sin75°≈ 9.66 N·m
TORQUE Homework Solution
![Page 82: The Cross Product - khu.ac.krweb.khu.ac.kr/.../BME101_05_2_Biomechancis_Cross_Product.pdfTHE CROSS PRODUCT The cross producta xb of two vectors aand b, unlike the dot product, is a](https://reader034.fdocuments.net/reader034/viewer/2022050206/5f59c1b492d08d42da3bb590/html5/thumbnails/82.jpg)
If the bolt is right-threaded, then the torque vector itself is
τ = |τ| n ≈ 9.66 n
where n is a unit vector directed down into the slide.
TORQUE Homework Solution