Download - Vectors in the Plane Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 A ball flies through the air at a certain speed.

Vectors in the Plane

Digital Lesson

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2

A ball flies through the air at a certain speed and in a particular direction. The speed and direction are the velocity of the ball. The velocity is a vector quantity since it has both a magnitude and a direction.

Vectors are used to represent velocity, force, tension, and many other quantities.

A vector is a quantity with both a magnitude and a direction.


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

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

u

v

The vector v = PQ is the set of all directed line segments

of length ||PQ|| which are parallel to PQ.

P

Q


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

For example, the scalar 3 times v results in the vector 3v, three times as long and in the same direction as v.

v

3v

v

The product of - and v gives a vector half as long

as and in the opposite direction to v. 2

1

2

1- v


Vector Addition

To add vectors u and v:

1. Place the initial point of v at the terminal point of u.

2. Draw the vector with the same initial point as u and the same terminal point as v.

uv

u + v

v u

vu


Vector Subtraction

To subtract vectors u and v:

1. Place the initial point of v at the initial point of u.

2. Draw the vector u v from the terminal point of v to the terminal point of u.

vu

v

u

v

u

u v


A vector with initial point (0, 0) is in standard position and is represented uniquely by its terminal point (u1, u2).

If v is a vector with initial point P = (p1 , p2) and terminal point Q = (q1 , q2), then

1. The component form of v is

v = q1 p1, q2 p2

2. The magnitude (or length) of v is

||v|| =2

222

11 )()( pqpq

x

y(u1, u2)

x

y

P (p1, p2)Q (q1, q2)


Example:

Find the component form and magnitude of the vector v with initial point P = (3, 2) and terminal point Q = (1, 1).

The magnitude of v is

||v|| = = = 5.2522 )3()4(

= , 34

p1 , p2 = 3, 2

q1 , q2 = 1, 1

So, v1 = 1 3 = 4 and v2 = 1 ( 2) = 3.

Therefore, the component form of v is , v2v1


Example: If u = PQ, v = RS, and w = TU with P = (1, 2),

Q = (4, 3), R = (1, 1), S = (3, 2), T = (-1, -2), and U = (1, -1),

determine which of u, v, and w are equal.

Calculate the component form for each vector:

u = 4 1, 3 2 = 3, 1

v = 3 1, 2 1 = 2, 1

w = 1 (-1), 1 (-2) = 2, 1

Therefore v = w but v = u and w = u./ /

Two vectors u = u1, u2 and v = v1, v2 are equal if and only if u1 = v1 and u2 = v2 .


Operations on Vectors in the Coordinate Plane

Let u = (x1, y1), v = (x2, y2), and let c be a scalar.

1. Scalar multiplication cu = (cx1, cy1)

2. Addition u + v = (x1+x2, y1+ y2)

3. Subtraction u v = (x1 x2, y1 y2)


x

y

u + v

x

y

Examples: Given vectors u = (4, 2) and v = (2, 5)

-2u = -2(4, 2) = (-8, -4)

u + v = (4, 2) + (2, 5) = (6, 7) u v = (4, 2) (2, 5) = (2, -3)

(4, 2)u

(-8, -4) 2u

x

y

(6, 7)(2, 5)

(4, 2)v

u

(2, 5)

(4, 2)v

u

(2, -3)

u v


x

y

The direction angle of a vector v is the angle formed by the positive half of the x-axis and the ray along which v lies.

x

y

vθ v

θ

x

y

v

x

y

(x, y)

If v = 3, 4 , then tan = and = 51.13.

3

4

If v = x, y , then tan = . x

y