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6.2 - Dot Product of Vectors

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 = <u 1 ,u 2 > and v = <v 1 ,v 2 > is u • v = u 1 v 1 + u 2 v 2 Used to calculate the angle between two vectors. Properties of the Dot Product.

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6.2 - Dot Product of Vectors

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

  2. 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

  3. 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)

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

  5. Use the dot product to find the length of the vector • u = <4,-3>

  6. 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|))

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

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

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

  10. 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

  11. 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

  12. 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.

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

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