Vector Geometry Operations

Addition and Subtraction

If ${\displaystyle \mathbf {a} \,}$ and ${\displaystyle \mathbf {b} \,}$ are vectors,

Then the sum

${\displaystyle \mathbf {c} =\mathbf {a} +\mathbf {b} \,}$ is also a vector (see Figure 2(a)).

The two vectors can also be subtracted from one another to give another vector

${\displaystyle \mathbf {d} =\mathbf {a} -\mathbf {b} \,}$.

Multiplication by a scalar

Multiplication of a vector ${\displaystyle \mathbf {b} \,}$ by a scalar ${\displaystyle \lambda \,}$ has the effect of stretching or shrinking the vector (see Figure 2(b)).

You can form a unit vector ${\displaystyle {\mathbf {\hat {b}} }\,}$ that is parallel to ${\displaystyle \mathbf {b} \,}$ by dividing by the length of the vector ${\displaystyle |\mathbf {b} |\,}$. Thus,

${\displaystyle {\mathbf {\hat {b}} }={\cfrac {\mathbf {b} }{|\mathbf {b} |}}~.}$