Today in Precalculus Turn in graded wkst and

Today in Precalculus • Turn in graded wkst and page 511: 1 -8 • Notes: Dot Product Angle between Two Vectors Orthogonal & Parallel Vectors • Homework • Quiz Friday

Dot Product The dot product of u= u 1, u 2 and v= v 1, v 2 is u • v=u 1 v 1 + u 2 v 2 Examples: Find each dot product: 4, -1 • 8, 3 = 32 + -3 = 29 2, -3 • -4, -1 = -8 + 3 = -5 4, 2 • -3, 5 = -12 + 10 = -2

Properties of Dot Product Let u, v and w be vectors and let c be a scalar. 1. u • v=v • w 2. u • u=|u|2 3. 0 • u=0 4. u • (v+w)=u • v + v • w 5. (cu) • v=u • (cv) = c(u • v)

Using Properties of Dot Product Find the length of u= -2, 4 using dot product. u • u=|u|2 u • u= 4+16 = 20 So |u|2 = 20 Then |u| =

Angles Between Two Vectors If θ is the angle between two nonzero vectors u and v, then

Angles Between Two Vectors Find the angle between vectors u and v. u = 3, 5 v = -2, 1 u • v = -6 + 5 = -1

Angles Between Two Vectors Find the angle between vectors u and v. u = -1, -3 v = 2, 1 u • v = -2 + -3 = -5

Orthogonal Vectors If vectors u and v are perpendicular, then u • v = |u| |v|cos 90°=0 The vectors u and v are orthogonal, then u • v = 0 For non-zero vectors, orthogonal and perpendicular have the same meaning. Zero vectors have no direction angle, so they are not perpendicular to any vector. They are orthogonal to every vector. Ex: Prove u = 3, 2 and v = -8, 12 are orthogonal. u • v = -24 + 24 = 0

Parallel Vectors If vectors u and v are parallel iff: u = kv for some constant k. Ex: Prove u = 3, 2 and v = -6, -4 are parallel. -2 3, 2 = -6, -4

Proving Vectors are Neither If vectors u and v are not orthogonal or parallel, then they are neither. Show that vectors u and v are neither: u = 3, 2 v = -4, -6 u • v = -12 + -12 = -24 ≠ 0

Practice Find the dot product: 5, 3 • 12, 4 = 60 + 12 = 72 Use the dot product to find |u| if u = 5, -12 so |u| =13 Find the angle θ between u = -4, -3 and v = 1, 5

Homework Pg 519: 1 -4, 9, 10, 13 -16, 21, 22, 33 -38 Quiz Friday