Multiplication with Vectors

Scalar MultiplicationDot Product

Cross Product

Objectives

• TSW use the dot product to fin the relationship between two vectors.

• TSWBAT determine if two vectors are perpendicular

A bit of review

• A vector is a _________________

• The sum of two or more vectors is called the ___________________

• The length of a vector is the _____________

Find the sum

• Vector a = < 3, 9 > and vector b = < -1, 6 >

• Find

• What is the magnitude of the resultant.

• Hint* remember use the distance formula.

a b

Multiplication with Vectors

Scalar MultiplicationDot Product

Cross Product

Scalar Multiplication: returns a vector answer

Distributive Property:

If 4 6 find 2a i j a

If 3 2 b 5 find 2 3a i j i j a b

Multiplication with Vectors

Scalar MultiplicationDot ProductCross Product

Dot Product

• Given and are two vectors, • The Dot Product ( inner product )of and is defined

as

• A scalar quantity

1 2( , )a a a1 2( , )b b b

a b

1 1 2 2a b a b a b

Finding the angle between two Vectors

a

b

θa - b

2 22

22

22

2 22 2

2 cos( )

2 cos( )

2 2 cos( )

2 2 cos( )

cos( )

a b a b a b

a b a b a b a b

a a a b b b a b a b

a a b b a b a b

a ba b

Example• Find the angle between the vectors:

3,6 and 4, 2r s

1:

2:

3:

Angle between Vectors

cos( ) a ba b

Classify the angle between two vectors:

Acute : ______________________________________________

Obtuse: _____________________________________________

Right: (Perpendicular , Orthogonal) _______________________

example

• Given three vectors determine if any pair is perpendicular

3,12a 8, 2b 3,2 c

THEOREM: Two vectors are perpendicular iff their Dot (inner) product is zero.

Ex 1:

Ex 2:

Find the unit vector in the same direction as v = 2i-3j-6k

Ex 3:

Ex 4: If v = 2i - 3j + 6k and w = 5i + 3j – k

Find: ( ) (b) w v (c) v v

(d) w w (e) v (f) w

a v w

(c) 3v (d) 2v – 3w (e) v

Ex 5:

Ex 6: Find the angle between u = 2i -3j + 6k and v = 2i + 5j - k

Ex 7: Find the direction angles of

v = -3i + 2j - 6k

Ex 8: The vector v makes an angle of with the 3

positivex-axis, an angle of = with the positive y-axis,3

and an acute angle with the positive z-axis. Find .

Any nonzero vector v in space can be written in terms of its magnitude and direction cosines as:

Ex 9: Find the direction angles of the vector below. Write the answer in the form of an equation.

v = 3i – 5j + 2k 9 25 4 38V

5cos ; 14438

3cos ; 6138

2cos ; 7138

38 cos 61 cos144 cos 71i j k

• We can also find the Dot Product of two vectors in 3-d space.

• Two vectors in space are perpendicular iff their inner product is zero.

1 1 2 2 3 3a b a b a b a b

Example

• Find the Dot Product of vector v and w.

• Classify the angle between the vectors.

6,2,10 and 4,1,3v w

Projection of Vector a onto Vector b

a

ba

b

Written : bproj a

2 ba bproj a bb

Example:

Find the projection of vector a onto vector b :

3 and - 2 4a i j b i j

4,1, 1 and 2, 3, 3a b

Decompose a vector into orthogonal components…

Find the projection of a onto b

Subtract the projection from a

The projection, and a - b are orthogonal

a

b

a-b bproj a

bproj a

Multiplication with Vectors

Scalar MultiplicationDot Product

Cross Product

OBJECTIVE 1

find v w

OBJECTIVE 2

OBJECTIVE 3

OBJECTIVE 4

OBJECTIVE 5

Cross product• Another important product for vectors in

space is the cross product. • The cross product of two vectors is a vector.

This vector does not lie in the plane of the given vectors, but is perpendicular to each of them.

• If

1 2 3

1 2 3

, ,

, ,

a a a a

b b b b

Then the cross product of vector a and vector b is defined as follows:

2 3 1 3 1 2

2 3 1 3 1 2

a a a a a aa b i j k

b b b b b b

The determinant of a 2 x 2 matrix

1 11 2 1 2

2 2

= - a b

a b b aa b

2 5 =

3 4

• An easy way to remember the coefficients of vectors I, j, and k is to set up a determinant as shown and expand by minors using the first row.

1 2 3

1 2 3

i j ka a ab b b

You can check your answer by using the dot product.

2 3 1 3 1 2

2 3 1 3 1 2

a a a a a ai j k

b b b b b b

Example• Find the cross product of vector a and vector b

if: 5,2,3 2,5,0a b

Verify that your answer is correct.

Assignment