MAT 1236Calculus III

Section 12.3 The Dot Product

http://myhome.spu.edu/lauw

HW… WebAssign 12.3 (19 problems, 108 min.)

Preview Define a new operation on vectors. Angle(s) between vectors Projection of vectors

• Formula• Why the formula makes sense? (More

important in a long run)

We are Interested in … The angle between two forces The component of one force along the

direction of another force

Vectors The angle between two forces

Dot ProductIf and , the dot product of a and b is the number

1 2 3 1 2 3

1 1 2 2 3 3

, , , ,a b a a a b b b

a b a b a b

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

Example 1(a)

(b)

1,0, 1 0,1, 1

2 3i j k i k

Dot product and Length

1 2 3 1 2 3, , , ,a a a a a a a a

Properties

Properties

Geometric Meaning0

a

b a b

a

b a b

2v v v

Formula0

a

b

cos a ba b

PPFNE Derive the formula of the angle between

two vectors

Example 2Find the angle between the given vectors

1, 1,1 , 1, 1,2a b

cos a ba b

Special Case: Orthogonal VectorsTwo vectors are orthogonal if the angle between them is a right angle.

cos 0a ba b

Example 3Find the value of x such that the given vectors are orthogonal.

, , 1 , 1, ,6a x x b x

ProjectionVector Projection

Scalar Projection

2aa bproj b aa

a

b

aproj b

acomp b

aa bcomp ba

Remark

Scalar Projection

aa bcomp ba

a

b

aproj b

acomp b

2

Make Sense?Vector Projection

How do I know the vector is in the same direction of a?

2aa bproj b aa

a

b

aproj b

acomp b

02

Make Sense?Vector Projection

How do I know <2,2> is in the same direction of <1,1>?

2aa bproj b aa

a

b

aproj b

acomp b

02

Make Sense?Vector Projection

Does the “length” of the vector agree with what we know?

2aa bproj b aa

aa bcomp ba

a

b

aproj b

acomp b

02

Proof: We need to recall…Vector Projection

2aa bproj b aa

a

b

aproj b

acomp b

Unit Vectors A unit vector is a vector with length is 1 e.g.

Unit VectorsThe unit vector in the same direction as vector

u

b

Unit VectorsUnit vector in the same direction as vector

So,

A vector equals to its length times the unit vector along the same direction

u

b

Proof:

a

b

x

Example 4Find the vector projection of onto .

2,3 , 4,1a b

2aa bproj b aa