1. Precalculus
8-4 Perpendicular Vectors
Inner Product
The inner product of two vectors results in a _____________________, not a
_____________________.
• Inner product of vectors in a plane ( ___________________________________ )
r r r ρ
For a = α1 , α2 and b = β1 , β2 , the inner product a ⋅ β = _________________________
Two vectors are ___________________________ ⇔ their inner product is __________.
r r r
EX1: p = 7,14 , q = 2, −1 , and m = 3,5
Find the inner product. Are the vectors perpendicular.
r ρ r ρ r ρ
a) p⋅θ b) p⋅µ c) q⋅µ
• Inner product of r vectors in a space ( ___________________________________ )
r r ρ
For a = α1 , α2 , α3 and b = β1 , β2 , β3 , the inner product a ⋅ β = ____________________
r r r r r r
EX2: Find the inner product of a and b if a = −3,1,1 and b = 2, 8, −2 . Are a and b
perpendicular?

2. Cross Product
• The cross product of two vectors is a __________________________.
r
r
• The cross product of a and b is _______________________________ to both
r
r
a and b .
r r r r
If a = α1 , α2 , α3 and b = β1 , β2 , β3 , then the cross product of a and b is defined:
Remember how to find determinants?
r r r ρ
EX3: Find the cross product of v and w if v = 0, 3,1 ανδ ω = 0,1,2 . Verify that the
r r
resulting vector is perpendicular to both v and w .
HW p. 509 (12 – 30 even)