Section 10.2a

VECTORS IN THE PLANE

Vectors in the PlaneSome quantities only have magnitude, and are called scalars… Examples?Some quantities have both magnitude and direction, and canbe represented by directed line segments… Examples?

The directed line segment has initial point A andterminal point B; its length is denoted by . Directedline segments that have the same length and direction areequivalent.

AB��������������

AB��������������

A

BInitialpoint

TerminalpointAB

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Vectors in the PlaneDefinitions: Vector, Equal VectorsA vector in the plane is represented by a directed linesegment. Two vectors are equal (or the same) if they havethe same length and direction.

0,0

Definition: Component Form of a VectorIf v is a vector in the plane equal to the vector with initialpoint and terminal point , then thecomponent form of v is

1 2,v v

1 2,v vv

The magnitude (length) of v is 2 21 2v v v

Finding Component FormFind the (a) component form and (b) length of the vectorwith initial point P = (–3, 4) and terminal point Q = (–5, 2).

v 5 3 ,2 4 (a)

Start with a graph of the vector.

2, 2

2 2v 2 2 (b) 8 2 2

If a vector has a magnitude of 1, then it is a unit vector.The slope of a nonvertical vector is the slope shared by thelines parallel to the vector.

The zero vector: 0 is the only vector with nodirection.

0,0

Vectors in the PlaneDefinitions: Vector OperationsLet , be vectors with k a scalar(real number).

Addition:

1 2u ,u u 1 2v ,v v

1 1 2 2u v ,u v u v

Subtraction: 1 1 2 2u v ,u v u v

Scalar Multiplication: 1 2u ,k ku ku

Negative (opposite): 1 2u 1 u ,u u

Vectors in the PlaneVector AdditionWhen adding vectors geometrically, align them “head to tail,”and the sum is called the resultant vector. This geometricdescription of vector addition is sometimes called theparallelogram law:

uv

vu

u + v

Vectors in the PlaneScalar MultiplicationWhen multiplying scalars and vectors geometrically, thescalar simply stretches (k > 1) or shrinks (k < 1) the vector.If k is negative, the vector also changes to the oppositedirection:

u 2u

–2u0.7u

Vectors in the PlaneProperties of Vector OperationsLet u, v, and w be vectors and a, b be scalars.

1. u v v u 2. u v w u v w

3. u 0 u 4. u u 0

5. 0u 0 6. 1u u

7. u ua b ab 8. u v u va a a

9. u u ua b a b

Vectors in the PlaneDefinition: Dot Product (Inner Product)The dot product (or inner product) u v (“u dot v”) ofvectors and is the number1 2u ,u u 1 2v ,v v

1 1 2 2u v u v u v

Definition: Angle Between Two VectorsThe angle between nonzero vectors u and v is

1 u vcos

u v

Practice ProblemsFind the component form of the vector v of length 3 thatmakes an angle of with the positive x-axis.137v 3cos137 ,3sin137 2.194,2.046

Let and . Find:u 1,1 v 2, 5

4u v 4 1,1 2, 5 6,9

2u 3v 2 1,1 3 2, 5 8, 17

228 17 353

Practice ProblemsFind the measure of angle C in the triangle ABC defined bythe following points: 0,0A 3,5B 5,2C

Sketch a graph of the triangle.

The angle is formed by vectors and .CA��������������

CB��������������

Component forms of these vectors:

0 5,0 2CA ��������������

5, 2

3 5,5 2CB ��������������

2,3 1cosCA CB

m CCA CB

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����������������������������Angle between these vectors:

Practice ProblemsFind the measure of angle C in the triangle ABC defined bythe following points: 0,0A 3,5B 5,2C

1cosCA CB

m CCA CB

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5, 2CA ��������������

2,3CB ��������������

1

2 2 2 2

5 2 2 3cos

5 2 2 3

Practice ProblemsFind the measure of angle C in the triangle ABC defined bythe following points: 0,0A 3,5B 5,2C

5, 2CA ��������������

2,3CB ��������������

1

2 2 2 2

5 2 2 3cos

5 2 2 3m C

1 4

cos29 13

78.111 1.363

Practice ProblemsFind a unit vector in the direction of the given vector.

4, 3 Since a unit vector has a magnitude of 1, simplydivide the given vector by its own magnitude(this will keep the direction the same but stretchor shrink the vector to the correct length).

22

4, 3u

4 3

4, 3

16 9

Unit Vector:4, 3

5

4 3,5 5

How can we verify this answer?