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The Dot Product Angles Between Vectors Orthogonal Vectors

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The dot product or inner product of u = u , u

and v = v , v is

1

2

1

2

u v = u v + u v

1

1

2

2

vector!

The sum of two vectors is a…

vector!

The product of a scalar and a vector is a…

scalar!

The dot product of two vectors is a…

Let u, v, and w be vectors and let c be a scalar.

1. u v = v u

2

2. u u = |u|

3. 0 u = 0

4. u (v + w) = uv + uw

(u + v) w = uw + vw

5. (cu) v = u (cv) = c(uv)

Definition: Orthogonal Vectors

The vectors u and v are orthogonal if

and only if u v = 0.

- The terms “orthogonal” and “perpendicular”

are nearly synonymous (with the exception

of the zero vector)

Use the dot product to find the length of

vector v = 4, –3

(hint: use property 2!!!)

Length = 5

Find the angle between vectors u and v

Prove that the vectors u = 2, 3 and

v = –6, 4 are orthogonal

Check the dot product:

u v = 0!!!

Graphical Support???

First, let’s look at a brain exercise…

Page 520, #30:

Find the interior angles of the triangle with vertices (–4,1),

(1,–6), and (5,–1).

Start with a graph…

A(–4,1)

B(5,–1)

C(1,–6)

The vector projection of u = PQ onto a nonzero vector v = PS

is the vector PR determined by dropping a perpendicular from

Q to the line PS.

Q

u

P

S

v

R

Thus, u can be broken into components PR and RQ:

u = PR + RQ

Q

u

P

S

R

Notation for PR, the vector projection of u onto v:

PR = proj u

v

The formula:

u v

proj u = v

v

2

|v|

Find the vector projection of u = 6, 2 onto v = 5, –5 .

Then write u as the sum of two orthogonal vectors, one

of which is proj u.

v

Start with a graph…

Find the vector projection of u = 6, 2 onto v = 5, –5 .

Then write u as the sum of two orthogonal vectors, one

of which is proj u.

v

Start with a graph…

u = proj u + u = 2, –2 + 4, 4

v

2

Find the vector projection of u = 3, –7 onto v = –2, –6 .

Then write u as the sum of two orthogonal vectors, one

of which is proj u.

v

u = proj u + u = –1.8,–5.4 + 4.8,–1.6

v

2

Find the vector v that satisfies the given conditions:

2

u = –2,5 , u v = –11, |v| = 10

A system to solve!!!

Find the vector v that satisfies the given conditions:

2

u = –2,5 , u v = –11, |v| = 10

OR

Now, let’s look at p.520: 34-38 even:

What’s the plan???

If u v = 0 orthogonal!

If u = kv parallel!

34) Neither

36) Orthogonal

38) Parallel

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