Dot Product/Examples

Examples of Dot Product

Let:

$\ds \mathbf A$

$=$

$\ds 6 \mathbf i + 4 \mathbf j + 3 \mathbf k$

$\ds \mathbf B$

$=$

$\ds 2 \mathbf i - 3 \mathbf j - 3 \mathbf k$

Then:

$\mathbf A \cdot \mathbf B = -9$

Hence the angle between $\mathbf A$ and $\mathbf B$ is approximately $104.2 \degrees$.

Let $\mathbf F$ represent a force acting on a body $B$.

Let $\mathbf d$ denote the displacement effected on $B$ by $\mathbf F$.

Then the work done by $\mathbf F$ on $B$ is given by:

$W = \mathbf F \cdot \mathbf d = \norm {\mathbf F} \norm {\mathbf d} \cos \theta$

where:

$\cdot$ denotes dot product

$\theta$ is the angle between the directions of $\mathbf F$ and $\mathbf d$.