Precalculus – Vectors – Dot Product
Precalculus – Vectors – Dot Product
The dot product of u= (u1,u2( and v= (v1,v2( is
Examples: Find each dot product:
(4,-1(•(8,3( =
(2,-3(•(-4,-1( =
(4,2(•(-3,5( =
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.
Angle Between Two Vectors
Find the angle between vectors u and v. Where u = (3,5( and v = (-2,1(
Find the angle between vectors u and v. Where u = (-1,-3( and v = (2,1(
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.
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
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(
Practice
Find the dot product: (5,3(•(12,4(
Use the dot product to find |u| if u = (5, -12(
Find the angle θ between u = (-4,-3( and v = (-1,5(
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