Calculus II/Vector Geometry Operation

Addition and Subtraction

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

Then the sum

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

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

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

Multiplication by a scalar

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

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

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