8
2.3
Vectors
Formulas for vector products
Some formulas for dot and cross products and their combinations p×q
=
−(q × p)
p · (q × r)
=
(p × q) · r
(p × q) · (p × q)
=
(p · p)(q · q) − (p · q)2
p × (q × r)
=
q(p · r) − r(p · q)
Example 2.1. What are the dot products of combinations of these vectors? −1 2 1 a= , b= , c= 4 1/2 −1 Answer: a · b = (−1) · 2 + 4 · (1/2) = 0 a · c = (−1) · 1 + 4 · (−1) = −5 b · c = 2 · 1 + (1/2) · (−1) = 1.5 so only the pair a and b are orthogonal. Example 2.2. What are the length of the vectors a, b, and c in the previous example? a·a=
|a|2 =
(−1) · (−1) + 4 · 4 = 17
2
b · b = |b| = c·c=
2 · 2 + (1/2) · (1/2) = 4.25
2
|c| =
so the lengths of a, b, and c are
√
1 · 1 + (−1) · (−1) = 2 √ √ 17, 4.25 and 2.
Example 2.3. If the two vectors 1 0 a = 0 , b = 1 , 0 0
then
0 a × b = 0 1