Vectors in the planeVector operations

Vectors

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

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

x

y(u1, u2)

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

P

Q

VECTORS x

y(u1, u2)

4

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|| =

x

y(u1, u2)

x

y

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

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

= , 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

The magnitude of v is

||v|| = = = 5.

Find the component form and magnitude of the vector v with initial point P = (4, −7) and terminal point Q = (−1, 5).

Try this on your own!

||v||=13

Vector Properties

1. u + v = v + u

3. u + 0 = u

5. c(du) = (cd)u

7. c(u+v) = cu + cv

9. ||cv|| = |c| ||v||

2. (u + v) + w = u + (v + w)

4. u + (-u) = 0

6. (c+d)u = cu + du

8. (c-d)u = cu - du

10. 1(u) = u, 0(u) = 0

Warm - upLet v= <-2,5> and w=<3,4>

FIND: a. 2v b. w-v c. v+2wa. <-4,10> b. <5,-1> c. <4,13>Let v= <-4,2> and w=<1,5>

FIND: a. 2v b. w-v c. v+2w

Unit vector

A unit vector is a vector of unit length

Find the unit vector in the direction of v = <-2,5>