Precalculus Notes Lesson 6.3 Vectors in the Plane Part 1

A vector is a quantity with both a magnitude and a direction. A quantity that does not involve direction is

called a scalar. For example, 55 miles per hour is a scalar but 55 miles per hour heading north is a vector.

Symbols for vectors: A, A, ⃗⃗ , ⃗⃗

The magnitude of a vector is its length. It can be found using the distance formula.

|| ⃗⃗⃗⃗⃗⃗ || = √

A quantity with magnitude and direction is represented by a directed line segment PQ with initial point P and

terminal point Q.

The vector v = ⃗⃗⃗⃗ ⃗ is the set of all directed line segments of magnitude ||PQ|| which are parallel to ⃗⃗⃗⃗ ⃗.

Two vectors, u and v, are equal if the line segments representing them are parallel and have the same length or

magnitude.

Example 1: Let u be represented by the directed line segment from P(0, 0) to Q(3, 1) and let v be represented by

the directed line segment from R = (2, 2) to S = (5, 3).

a. Sketch u and v. b. Show that u and v are equivalent.

Scalar multiplication is the product of a scalar, or real number, times a vector.

Example 2: a. 2w b. ½ w c. – w

Q

P

w

v

Q

Vector Addition (Geometric) v + u

To add two vectors: 1. Place the tail (initial point) of one vector at

the tip (terminal point) of the other vector. 2. Connect the exposed tail to the exposed tip.

Vector Subtraction (Geometric) v – u = v + –u

To subtract two vectors:

Change the problem to addition by adding

the opposite. Then add the two vectors.

A vector with initial point (0, 0) is in standard position and is

represented uniquely by its terminal point ( , ). The component form of this vector is written as v = ⟨ ⟩

Example 3:

a) Find the component form of the vector v with initial point P = (3, 4) and terminal point Q = (1, 1).

b) Sketch ⃗⃗⃗⃗ ⃗ and the component vector v.

c) Find the magnitude of v.

Let v be a vector with initial point P = (p1 , p2) and terminal point Q = (q1 , q2)

The component form of v is

v = ⟨ ⟩ ⟨q1 p1, q2 p2 ⟩ = ⃗⃗ ⃗⃗ ⃗

The magnitude (or length) of v is

||v|| = √

= √

If ||v|| = 1, then v is a unit vector. If ||v|| = 0, then v is the zero vector 0.

u

v

P

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Operations on Vectors in the Coordinate Plane (Algebraic)

Let u = ⟨ ⟩, v = ⟨ ⟩, and let k be a scalar.

1. Scalar multiplication ku = ⟨kx1, ky1⟩

2. Addition u + v = ⟨x1 + x2, y1 + y2⟩

3. Subtraction u v = ⟨x1 x2, y1 y2⟩

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Example 4: Let u = ⟨ ⟩ and v = ⟨ ⟩. Find each of the following vectors. Sketch the resultant vector.

a. –2u b. u + v

c. u – v d. 2u – 3v

Properties of Vector Addition and Scalar Multiplication

Let u, v, and w be vectors and let c and d be scalars. Then the following properties are true.

1. u + v = v + u 2. (u + v) + w = u + (v + w) 3. u + 0 = u

4. u + (–u) = 0 5. c(du) = (cd)u 6. (c + d)u = cu + du

7. c(u + v) = cu + cv 8. 1(u) = u, 0(u) = 0 9. ||cv|| = |c| · ||v||

Homework: Page 454 -457 # 1 – 8, 11 – 23 odd, 31 – 34 all, 103 – 110

31. 32.

33. 34.