DOT PRODUCTS AND PROJECTIONS MATH 200 MAIN QUESTIONS FOR TODAY - - PowerPoint PPT Presentation

DOT PRODUCTS AND PROJECTIONS

MATH 200 WEEK 1- FRIDAY

MATH 200

MAIN QUESTIONS FOR TODAY

▸ How is the dot product defined for vectors? ▸ How does it interact with other operations on vectors? ▸ What uses are there for the dot product?

MATH 200

DEFINITION

▸ The dot product is a new kind of operation in that it takes

in two objects of one kind and yields an object of a different kind!

▸ It takes two vectors and gives a scalar ▸ Given v = <v1, v2, v3> and w = <w1, w2, w3>, we define the

dot product as follows

▸ v • w = v1w1 + v2w2 + v3w3 ▸ E.g. If v = <2, 1, -2> and w = <3, -4, -1>, then ▸ v • w = (2)(3) + (1)(-4) + (-2)(-1) = 6 - 4 + 2 = 4

MATH 200

EXAMPLES

▸ Compute the following dot products: h1, 4, 5i · h2, 2, 1i = (1)(2) + (4)(2) + (5)(1) = 2 + 8 + 5 = 15

MATH 200

PROPERTIES OF THE DOT PRODUCT

▸ The dot product is called a product because of how it

interacts with vector addition:

~ a · (~ v + ~ w) = ~ a · ~ v + ~ a · ~ w

▸ It’s commutative (meaning the order in which we multiply

doesn’t matter):

▸ And it can be used to define the norm of a vector more

succinctly:

~ v · ~ v = ||~ v||2

~ v · ~ w = ~ w · ~ v

***For each property, you should confirm with examples***

MATH 200

WHAT DOES THIS DO FOR US?

▸ Remember of the Law of Cosines…? ▸ Of course you do - it’s a generalized Pythagorean Theorem

a b c θ

c2 = a2 + b2 − 2ab cos θ

MATH 200

▸ Let’s redraw the law of cosines diagram with vectors

instead:

a b c θ

v w v-w

c2 = a2 + b2 − 2ab cos θ WHICH OPERATION ON V AND W GIVES US THE REMAINING SIDE?

||~ v − ~ w||2 = ||~ v||2 + ||~ w||2 − 2||~ v||||~ w|| cos ✓

||~ v − ~ w||2 = ||~ v||2 + ||~ w||2 − 2||~ v||||~ w|| cos ✓

||~ v − ~ w||2 = (~ v − ~ w) · (~ v − ~ w) = ~ v · ~ v − 2~ v · ~ w + ~ w · ~ w = ||~ v||2 − 2~ v · ~ w + ||~ w||2

EXPAND THIS TERM PLUG BACK IN

||~ v||2 − 2~ v · ~ w + ||~ w||2 = ||~ v||2 + ||~ w||2 − 2||~ v||||~ w|| cos ✓ −2~ v · ~ w = −2||~ v||||~ w|| cos ✓ ~ v · ~ w = ||~ v||||~ w|| cos ✓ cos ✓ = ~ v · ~ w ||~ v||||~ w||

MATH 200

QUICK CONCLUSIONS FROM THE DOT PRODUCT

▸ Say we compute the dot product of two vectors v and w.

The result will be positive, negative, or zero.

▸ What can we say about the angle between the vectors in

each case?

▸ If v • w > 0: cosθ > 0 so the angle is acute ▸ If v • w < 0: cosθ < 0 so the angle is obtuse ▸ If v • w = 0: cosθ = 0 so the angle is 90o ▸ We use the word orthogonal to refer to vectors that

form a 90o angle.

cos ✓ = ~ v · ~ w ||~ v||||~ w||

Reminder

MATH 200

PROJECTIONS

▸ Say we have two vectors v and b,

and we want to do the following:

▸ Draw v and b tail to tail ▸ For the sake of this

illustration make b longer than v though it doesn’t matter

▸ Drop a line that’s

perpendicular to b from the tip of v

▸ Find the vectors that form the

right triangle that results

b v

THIS VECTOR IS CALLED THE PROJECTION OF V ONTO B

MATH 200

▸ We write the projection of v onto b as projbv ▸ From the picture it should be clear that ▸ b/||b|| is a unit vector in the direction of b so…

b v

θ

MATH 200

▸ Putting it all together…

REMEMBER FROM BEFORE

X X

b v

θ

MATH 200

DISTANCE FROM A POINT TO A LINE

▸ Let’s use projections to find the distance from a point to a line. ▸ Find the (shortest) distance from the point A(3,1,-1) to the

line containing P1(6,3,0) and P2(0,3,3)

▸ We’re all about vectors now so let’s draw some…

A P1 P2

v b