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![Page 1: Vector Practice Questions. Question 1 Determine the magnitude and direction of the sum of two vectors u and v, if their magnitudes are 5 and 8 units respectively,](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649dc55503460f94ab92fe/html5/thumbnails/1.jpg)
Vector
Practice Questions
![Page 2: Vector Practice Questions. Question 1 Determine the magnitude and direction of the sum of two vectors u and v, if their magnitudes are 5 and 8 units respectively,](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649dc55503460f94ab92fe/html5/thumbnails/2.jpg)
Question 1Determine the magnitude and direction of the sum of two vectors u and v, if their magnitudes are 5 and 8 units respectively, and the angle between them is 30o
30
5u
8v
150
2 2 22 cos 150u v u v u v
2 25 8 2 5 8 cos 150
158.28
12.6u v
sin sin 150
5 12.65sin 150
s
11.4
in12.6
![Page 3: Vector Practice Questions. Question 1 Determine the magnitude and direction of the sum of two vectors u and v, if their magnitudes are 5 and 8 units respectively,](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649dc55503460f94ab92fe/html5/thumbnails/3.jpg)
Question 2Cassiopia is towing her friend on a toboggan using a rope which makes an angle of 25o with the ground. If Cassiopia is pulling with a force of 70N, what horizontal force is she exerting on the toboggan?
70cos 25h
63.4N
25h
![Page 4: Vector Practice Questions. Question 1 Determine the magnitude and direction of the sum of two vectors u and v, if their magnitudes are 5 and 8 units respectively,](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649dc55503460f94ab92fe/html5/thumbnails/4.jpg)
Question 3Alice and Bob are towing their friends on a toboggan. Each is exerting a horizontal force of 60N. Since they are walking side by side, the ropes pull one to each side; they each make an angle of 20o with the line of motion. Determine the force pulling the toboggan forward.
60N
60N
140
60cos 20 60cos 20r
112.8N
2020
![Page 5: Vector Practice Questions. Question 1 Determine the magnitude and direction of the sum of two vectors u and v, if their magnitudes are 5 and 8 units respectively,](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649dc55503460f94ab92fe/html5/thumbnails/5.jpg)
Question 4
If and3 2 4 2 2 2 5 find u i j k l v i j k l u v
2 2 2 2
2 5 2 3 2 5 4 2 2
2 1,3, 1,2 5 4, 1,2, 2
2,6, 2,4 20, 5,10, 10
22,11, 12,14
22 11 12 14
945
3 105
u v i j k l i j k l
![Page 6: Vector Practice Questions. Question 1 Determine the magnitude and direction of the sum of two vectors u and v, if their magnitudes are 5 and 8 units respectively,](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649dc55503460f94ab92fe/html5/thumbnails/6.jpg)
Question
Determine a unit vector for u=(1,2,3)
2 2 2
ˆ
1,2,3
1 2 3
1,2,3
14
1 2 3, ,
14 14 14
uu
u
![Page 7: Vector Practice Questions. Question 1 Determine the magnitude and direction of the sum of two vectors u and v, if their magnitudes are 5 and 8 units respectively,](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649dc55503460f94ab92fe/html5/thumbnails/7.jpg)
Question 5
Determine the angle between a=(1, -4) and b=(2, 5).
1
1
2 2 22
cos
cos
cos
1 2 4 5cos
1 4 2 5
144.2
a b a b
a b
a b
a b
a b
![Page 8: Vector Practice Questions. Question 1 Determine the magnitude and direction of the sum of two vectors u and v, if their magnitudes are 5 and 8 units respectively,](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649dc55503460f94ab92fe/html5/thumbnails/8.jpg)
Question
Find a vector that is perpendicular to both a=(1, -4,5) and b=(2, 5, -3).
1, 4,5 2,5, 3
13,13,13
v a b
How about a unit vector?
2 2 2
ˆ
13,13,13
13 13 13
13 13 13, ,
507 507 507
vv
v
![Page 9: Vector Practice Questions. Question 1 Determine the magnitude and direction of the sum of two vectors u and v, if their magnitudes are 5 and 8 units respectively,](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649dc55503460f94ab92fe/html5/thumbnails/9.jpg)
Find the direction cosines and direction angles of the vector
The magnitude of u
Question
0,5, 3u
2 2 20 5 3
34
u
0cos
345
cos343
cos34
90
31
121
Because vector is not used to point in the direction of the line, we have no requirement that the ‘z’ term be positive.
![Page 10: Vector Practice Questions. Question 1 Determine the magnitude and direction of the sum of two vectors u and v, if their magnitudes are 5 and 8 units respectively,](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649dc55503460f94ab92fe/html5/thumbnails/10.jpg)
Determine the angle, to the nearest degree, that (–4,1,5) makes with the positive x–axis.
Question
2 2 2
4
4 1 5cos
4
42128.1
![Page 11: Vector Practice Questions. Question 1 Determine the magnitude and direction of the sum of two vectors u and v, if their magnitudes are 5 and 8 units respectively,](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649dc55503460f94ab92fe/html5/thumbnails/11.jpg)
Question A vector makes angles of and , respectively, with the - and - axes.
What is the angle between and the -ax
60 105 x
is?
y
z
u
u
2 2 2
2 2
cos cos cos 1
cos 1 cos cos
60 105
60 1
34 1
0
46
5
or
![Page 12: Vector Practice Questions. Question 1 Determine the magnitude and direction of the sum of two vectors u and v, if their magnitudes are 5 and 8 units respectively,](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649dc55503460f94ab92fe/html5/thumbnails/12.jpg)
Determine the coordinates of the point that divides AB in the ratio 4:2 where A is (2, -1, 4) and B is (3, 1, 7)
n mOP OA OB
m n m n
������������������������������������������If AP:PB = m:n then
2 42, 1,4 3,1,7
4 2 4 21 2
2, 1,4 3,1,73 38 1 18
, ,3 3 3
OP
��������������
Question
![Page 13: Vector Practice Questions. Question 1 Determine the magnitude and direction of the sum of two vectors u and v, if their magnitudes are 5 and 8 units respectively,](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649dc55503460f94ab92fe/html5/thumbnails/13.jpg)
UnderstandingConsider collinear points P, Q, and R. consider also a reference point O. Write OQ as a linear combination of OP and OR if:
a) Q divides PR (externally) in the ratio -3:8
Q P R
0
-3 5
3
5PQ PR
����������������������������
OQ OP PQ ������������������������������������������
3
5OQ OP PR ������������������������������������������
3
5OQ OP PO OR ��������������������������������������������������������
3
5OQ OP OP OR ��������������������������������������������������������
3 3
5 5OQ OP OP OR ��������������������������������������������������������
8 3
5 5OQ OP OR ������������������������������������������
Since:
Then:
8
![Page 14: Vector Practice Questions. Question 1 Determine the magnitude and direction of the sum of two vectors u and v, if their magnitudes are 5 and 8 units respectively,](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649dc55503460f94ab92fe/html5/thumbnails/14.jpg)
Question
Find the projection of on 1,3,5 1,3, 2u v
2
2Proj
1,3,5 1,3, 21,3, 2
1,3, 2
21,3, 2
141 3 2
, ,7 7 7
u v vu vo
vn
�
2
coscos
1
cos
Pr v
u vu
v
u v
v
u v
v
u v v
v v
u v v
v
oj u
Multiply by unit vector in direction of v.
![Page 15: Vector Practice Questions. Question 1 Determine the magnitude and direction of the sum of two vectors u and v, if their magnitudes are 5 and 8 units respectively,](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649dc55503460f94ab92fe/html5/thumbnails/15.jpg)
Question Find the point of intersection of: 1 3
3 1 2
13 1
2 1 3
x y z
x y z
Write in parametric form: 3
1
2 3
x s
y s
z s
2 13
1
3
x t
y t
z t
Therefore: 3 2 13
1 1
2 3 3
s t
s t
s t
9
7
s
t
27
8
21
x
y
z
Solve first two Check with third
![Page 16: Vector Practice Questions. Question 1 Determine the magnitude and direction of the sum of two vectors u and v, if their magnitudes are 5 and 8 units respectively,](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649dc55503460f94ab92fe/html5/thumbnails/16.jpg)
Determine the vector equation of a line that is parallel to 2x – 5y + 12 = 0 and has the same x–intercept as the line (x,y) = (2,-3) + t(3,-6).
Question
Direction vector : 5,2d
Point : ,0 2, 3 3, 6
1 1
2 2
1,0
2
x t
t x
Vector Equation: 1, ,0 5,2
2x y t
![Page 17: Vector Practice Questions. Question 1 Determine the magnitude and direction of the sum of two vectors u and v, if their magnitudes are 5 and 8 units respectively,](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649dc55503460f94ab92fe/html5/thumbnails/17.jpg)
Determine the scalar equation of the plane passing through the point (2,3,4) and parallel to the lines (x,y,z)=(1,1,1)+s(1,2,3) and (x,y,z)=(-2,4,5)+t(6,-1,2).
Question
Direction vector : 1 21,2,3 , 6, 1,2d d
Normal :
1 2
1, 2,3 6, 1, 2
7,16, 13
n d d
Scalar Equation:
7 16 13 0
7 2 16 3
7 16 1
13 4
3 0 0
0
1
0
1
x y z D
D
x
D
y z
![Page 18: Vector Practice Questions. Question 1 Determine the magnitude and direction of the sum of two vectors u and v, if their magnitudes are 5 and 8 units respectively,](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649dc55503460f94ab92fe/html5/thumbnails/18.jpg)
Question Determine the distance from the point P(1, -1, 2) to the plane 3x-2y+z=4
2 2 2
Ax By Cz Dd
A B C
2 22
3 2 1 4
3 2 1
3 14
1
2
4
1 1
![Page 19: Vector Practice Questions. Question 1 Determine the magnitude and direction of the sum of two vectors u and v, if their magnitudes are 5 and 8 units respectively,](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649dc55503460f94ab92fe/html5/thumbnails/19.jpg)
Question Determine the distance from the point P(2, -3, 2) to the line
0P A vd
v
��������������
1 1 1
3 2 4
x y z
First we need point on line: (1, -1, 1)
Now vector between two points:
1 2, 1 3 ,1 2 1,2, 1
2 2 2
1,2, 1 3,2,4
3 2 4
10,7,4
165
29
29
![Page 20: Vector Practice Questions. Question 1 Determine the magnitude and direction of the sum of two vectors u and v, if their magnitudes are 5 and 8 units respectively,](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649dc55503460f94ab92fe/html5/thumbnails/20.jpg)
Question Determine the distance between two parallel planes x-y+z-10=0 and x-y+z+4=0
2 2 2
Ax By Cz Dd
A B C
2 22
1 1 1 10
1
4 0 0
1 1
14 3
3
Find a point on one of the planes:
Put y=z=0 into x-y+z+4=0, and we obtain x=-4. Therefore a point is P(-4,0,0)
We now just need to determine the distance from P(-4,0,0) to x-y+z-10=0
![Page 21: Vector Practice Questions. Question 1 Determine the magnitude and direction of the sum of two vectors u and v, if their magnitudes are 5 and 8 units respectively,](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649dc55503460f94ab92fe/html5/thumbnails/21.jpg)
Question Determine whether the following points are collinear: A(3,1,0), B(1,0,2), C(5, 2, -2)
(1,0, 2) (3,1,0)
( 2, 1,2)
AB OB OA
������������������������������������������
Find direction vector AB and AC and see if they are parallel:
(5, 2, 2) (3,1,0)
(2,1, 2)
AC OC OA
������������������������������������������
Therefore AB=-AC, thus AB and AC are parallel
Since they both share the common point A, then A, B, C are collinear
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Question Determine the scalar equation of the plane perpendicular to plane x-y+z=3 containing the line (x,y,z)=(2,1,1)+t(2,1,3)
The normal to the plane x-y+z=3 is a vector on the required plane : (1, -1, 1)
(2,1,3) (1, 1,1) 4,1, 3
Another vector on the plane is the direction vector of the line : (2, 1, 3)
The normal of the required plane is then :
The equation of the plane is then: 4x+y-3z+D=0
Substitute in the point on the line (2, 1, 1): 4 32 1 1 0
6D
D
4 3 6 0x y z Therefore:
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Question Given a line :
12
3 5
x z ky
And a plane: 2x-y-z+10=0
3,1,5 2, 1, 1 0
Determine the value k such that the line cuts the plane: a) at one single pointb) At many pointsc) At no point:
a) The direction vector of line is perpendicular to normal of plane, so the line and plane are parallel, therefore no single solution.
b) Sub (1,2, k) into the plane: 2(1)-(2)-(k)+10=0 => k=10
c) If k does not equal 10, there is no solution
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Determine the area of the parallelogram below, using the cross product.
6
10
400
2
sin
6 10 sin 40
38.6 [ ]
area
units
u v
u v
Question
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2 3 4 1
1
2 2
x y z
x y z
x y z
Determine the point of intersection of the three planes:
Question
2 3 4 1
1 1 1 1
1 2 1 2
1 1 1 1
2 3 4 1
1 2 1 2
1 1 1 1
0 1 6 3
0 1 0 3
1 1 1 1
0 1 0 3
0 1 6 3
1 0 1 4
0 1 0 3
0 0 6 0
1 0 1 4
0 1 0 3
0 0 1 0
1 0 0
0 1 0
0 0 1
4
3
0
The point of intersection is: (-4,-3,0)