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

MATH 200 WEEK 1- FRIDAY DOT PRODUCTS AND PROJECTIONS

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 = <v 1 , v 2 , v 3 > and w = <w 1 , w 2 , w 3 >, we define the dot product as follows ▸ v • w = v 1 w 1 + v 2 w 2 + v 3 w 3 ▸ 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: h 1 , 4 , 5 i · h 2 , 2 , 1 i = (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 ) = ~ v + ~ ~ a · ~ a · ~ w ▸ It’s commutative (meaning the order in which we multiply doesn’t matter): ~ v · ~ w = ~ w · ~ v ▸ And it can be used to define the norm of a vector more succinctly: v || 2 v · ~ 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 c b θ a c 2 = a 2 + b 2 − 2 ab cos θ

MATH 200 ▸ Let’s redraw the law of cosines diagram with vectors instead: w WHICH OPERATION ON V AND W GIVES US THE c 2 = a 2 + b 2 − 2 ab cos θ REMAINING SIDE? c b v-w θ v a w || 2 = || ~ v || 2 + || ~ w || 2 − 2 || ~ || ~ v |||| ~ w || cos ✓ v − ~

w || 2 = || ~ v || 2 + || ~ w || 2 − 2 || ~ || ~ v |||| ~ w || cos ✓ v − ~ EXPAND THIS TERM w || 2 = ( ~ || ~ w ) · ( ~ w ) v − ~ v − ~ v − ~ = ~ v · ~ v − 2 ~ v · ~ w + ~ w · ~ w v || 2 − 2 ~ w || 2 = || ~ v · ~ w + || ~ PLUG BACK IN w || 2 = || ~ v || 2 + || ~ v || 2 − 2 ~ w || 2 − 2 || ~ || ~ v · ~ w + || ~ v |||| ~ w || cos ✓ v · ~ w = − 2 || ~ v |||| ~ w || cos ✓ − 2 ~ v · ~ w = || ~ v |||| ~ w || cos ✓ ~ v · ~ ~ w cos ✓ = || ~ 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: cos θ > 0 so the angle is acute Reminder v · ~ ~ ▸ If v • w < 0: cos θ < 0 so the angle is obtuse w cos ✓ = || ~ v |||| ~ w || ▸ If v • w = 0: cos θ = 0 so the angle is 90 o ▸ We use the word orthogonal to refer to vectors that form a 90 o angle.

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 v ▸ 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 b tip of v ▸ Find the vectors that form the THIS VECTOR IS CALLED THE right triangle that results PROJECTION OF V ONTO B

MATH 200 v θ b ▸ We write the projection of v onto b as proj b v ▸ From the picture it should be clear that ▸ b/||b|| is a unit vector in the direction of b so…

MATH 200 ▸ Putting it all together… v θ b REMEMBER FROM BEFORE X X

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 P 1 (6,3,0) and P 2 (0,3,3) ▸ We’re all about vectors now so let’s draw some… A v P 2 b P 1