Section 2.10 Summary
The Dot Product is a vector multiplication process defined by
The result is a scalar value, and the operation is commutative, so
The resulting value is the product \(\proj_\vec{A}\vec{B}\) with the magnitude of \(\vec{B}\text{.}\) The result is a signed value which can be positive or negative. A negative value indicates that the projection of \(\vec{A}\) onto \(\vec{B}\) and \(\vec{B}\) have opposite senses.
Dot products are used in mechanics to find vector projections and the component of one vector in a direction which is parallel to another.
The Cross Product is a vector multiplication process defined by
If \(\vec{A}\) and \(\vec{B}\) are in the \(xy\) plane, this evaluates to
The result is a vector mutually perpendicular to the first two with a sense determined by the right hand rule. The magnitude is the product of the perpendicular component of \(\vec{A}\) with the magnitude of \(\vec{B}\text{.}\) The operation is not commutative, in fact
Cross products are used in mechanics to find the moment of a force about about a point.
Both vector products are most useful when working in three dimensions, since simpler approaches are available for two-dimensional problems.