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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