Dot Products and projections Math 200 Week 1- Friday Dot Products and projections

Main Questions for Today 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:

Properties of the dot product Math 200 Properties of the dot product The dot product is called a product because of how it interacts with vector addition: 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: ***For each property, you should confirm with examples***

What does this do for us? c b θ a Remember of the Law of Cosines…? 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

Which operation on v and w gives us the remaining side? Math 200 Let’s redraw the law of cosines diagram with vectors instead: w Which operation on v and w gives us the remaining side? v-w a b c θ v

Expand this term Plug back in

Quick conclusions from the dot product 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. Reminder

this vector is called the projection of v onto b 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

We write the projection of v onto b as projbv Math 200 b v θ 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…

Math 200 Putting it all together… b v θ Remember from before X

Distance from a point to a line 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 v v P1 P2 b v θ b