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Published byRandolf Bishop Modified over 9 years ago
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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
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Dot Product The dot product of u= u1,u2 and v= v1,v2 is
u•v=u1v1 + u2v2 Examples: Find each dot product: 4,-1•8,3 = = 29 2,-3•-4,-1 = = -5 4,2•-3,5 = = -2
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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)
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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| =
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Angles Between Two Vectors
If θ is the angle between two nonzero vectors u and v, then
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Angles Between Two Vectors
Find the angle between vectors u and v. u = 3,5 v = -2,1 u•v = = -1
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Angles Between Two Vectors
Find the angle between vectors u and v. u = -1,-3 v = 2,1 u•v = = -5
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Orthogonal Vectors If vectors u and v are perpendicular, then
u•v = |u| |v|cos90°=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 = = 0
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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. -23,2 = -6,-4
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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 = = -24 ≠ 0
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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
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Homework Pg 519: 1-4,9,10,13-16,21,22,33-38 Quiz Friday
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