A vector may be multiplied by a scalar. Can two vectors may be multiplied together? Yes, one way is through the dot product: If a = < a ₁ , b ₁ > and b = < a ₂ , b ₂ >, then the dot product (or scalar product, or inner product) of a and b is given by a.b = a ₁ a ₂ + b ₁ b ₂.

The magnitude of the vector a can be computed in terms of by a.a = | a | ²

The angle between two vectors a and b can be computed by cos(theta) = a.b / | a || b |

Two vectors are orthogonal (perpindicular) if and only if the dot product vanishes, a.b =0.

The projection of a vector gives the component of a vector in a certain direction. The scalar projection of b onto a is given by

comp _a b = a.b / |a| and the vector projection of b onto a is given by proj_a = (comp _a b) a /|a|

In physics, Work is given by the inner product of a force with a displacement vector. (The component of force perpindicular, or orthogonal, to the displacement does not generate work!)

If a = < a₁, a₂ >, then the orthogonal complement of a is given by a ^perp = < -a₂, a₁ >.

Last modified Mon Sep 9 21:22:37 CDT 1996