6.2 - Dot Product of Vectors

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6.2 - Dot Product of Vectors HW: Pg. 519-520 #1-18e, 24

The Dot Product DEF: The dot product or inner product of u = <u1,u2> and v = <v1,v2> is u • v = u1v1 + u2v2 Used to calculate the angle between two vectors

Properties of the Dot Product Let u, v, and w be vectors and let c be a scalar u • v = v • u u • u = |u|2 0 • u = u • (v + w) = (cu) • v = u • (cv) = c(u • v)

Find each dot product <3,4> • <5,2> <1,-2> • <-4,3> (2i - j) •(3i - 5j)

Use the dot product to find the length of the vector

Angle Between Two Vectors If ө is the angle between the nonzero vectors u and v, then Cos = (u • v)/( |u| |v| ) And  = cos-1((u • v)/ (|u| |v|))

Find the angle between the vector u and v u = <2,3>, v = <-2,5> u = <2,1>, v = <-1,-3>

Orthogonal Vectors The vectors u and v are orthogonal if and only if u • v=0

Prove that the vectors are orthogonal u = <2,3> and v = <-6,4>

Projecting One Vector onto Another The vector projection of u = PQ onto a nonzero vector v = PS is the vector PR determined by dropping a perpendicular from Q to the line PS. u = PR + RQ PR and RQ are perpendicular The standard notation for PR = projvu

Projection of u onto v If u and v are nonzero vectors, the projection of u onto v is Projvu = ((u • v)/(|v|2))v

Decomposing a vector into perpendicular components Find the vector projection of u = <6,2> onto v = <5,-5>. Write u as the sum of two orthogonal vectors.

HW: Pg. 520 #25-28, 33-38