Unit 11.1–11.4: Vectors, Dot Product, Cross Product

What does the dot product tell you?

How much two vectors point in the same direction.

Dot product formula

u·v=a₁a₂+b₁b₂+c₁c₂

Geometric dot product

u·v=|u||v|cosθ

Angle between vectors

θ=cos⁻¹[(u·v)/(|u||v|)]

Perpendicular vectors test

u·v=0

Projection of u onto v

projᵥu=((u·v)/(v·v))v

Scalar component of u along v

compᵥu=(u·v)/|v|

Cross product magnitude

|u×v|=|u||v|sinθ

Cross product gives

A vector perpendicular to both input vectors.

Parallel vectors test

u×v=0

Area of parallelogram

|u×v|

Area of triangle

½|u×v|

Right-hand rule

Curl fingers from u to v; thumb points in direction of u×v.

How to recognize a dot product problem

Keywords: angle, projection, component, work, perpendicular.

How to recognize a cross product problem

Keywords: area, normal vector, perpendicular, plane.

Quick shortcut for perpendicular

Check dot product first.

Quick shortcut for parallel

Check scalar multiples first, then cross product.

Common mistake with cross products

Order matters: u×v=-(v×u).