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Section 2.10 Summary

The Dot Product is a vector multiplication process defined by

\begin{equation*} \vec{A} \cdot \vec{B} = A \; B \cos \theta = A_x B_x + A_y B_y \end{equation*}

The result is a scalar value, and the operation is commutative, so

\begin{equation*} \vec{A}\cdot\vec{B}=\vec{B}\cdot\vec{A} \end{equation*}

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

\begin{equation*} \vec{A} \times \vec{B}= A \; B \sin \theta \; \hat{\vec{u}}\text{.} \end{equation*}

If \(\vec{A}\) and \(\vec{B}\) are in the \(xy\) plane, this evaluates to

\begin{equation*} \vec{A} \times \vec{B}=(A_y B_x - A_x B_y) \khat\text{.} \end{equation*}

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

\begin{equation*} \vec{A}\times \vec{B}= \mathbf{-} \vec{B} \times \vec{A} \text{.} \end{equation*}

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.