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The Dot Product Angles Between Vectors Orthogonal Vectors. The beginning of Section 6.2a. Definition: Dot Product. 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….

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The Dot Product

Angles Between Vectors

Orthogonal Vectors

The beginning of Section 6.2a


Definition: Dot Product

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…


Properties of the Dot Product

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)


Finding the Angle Between

Two Vectors

v – u

v

u

0


Theorem:

Angle Between Two Vectors

v – u

If 0 is the angle between nonzero

vectors u and v, then

v

u

0

and


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)


Guided Practice

Find each dot product

= 23

1. 3, 4 5, 2

= –10

2. 1, –2 –4, 3

3. (2i – j) (3i – 5j)

= 11


Guided Practice

Use the dot product to find the length of

vector v = 4, –3

(hint: use property 2!!!)

Length = 5


Guided Practice

Find the angle between vectors u and v

u = 2, 3 , v = –2, 5

0 = 55.491


Guided Practice

Find the angle between vectors u and v


Guided Practice

Find the angle between vectors u and v

0 = 90

Is there an easier way to solve this???


Guided Practice

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)


First, let’s look at a brain exercise…

A(–4,1)

B(5,–1)

C(1,–6)


First, let’s look at a brain exercise…

A(–4,1)

B(5,–1)

C(1,–6)


First, let’s look at a brain exercise…

A(–4,1)

B(5,–1)

C(1,–6)


Definition: Vector Projection

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


Definition: Vector Projection

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|


Some Practice Problems

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…


Some Practice Problems…

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


Some Practice Problems…

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


Some Practice Problems…

Find the vector v that satisfies the given conditions:

2

u = –2,5 , u v = –11, |v| = 10

A system to solve!!!


Some Practice Problems…

Find the vector v that satisfies the given conditions:

2

u = –2,5 , u v = –11, |v| = 10

OR


Some Practice Problems…

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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