# How do I do vector subtraction?

Dec 30, 2014

Given two vectors $\vec{v}$ and $\vec{w}$ you have:
$\vec{v} - \vec{w} = \vec{v} + \left(- \vec{w}\right)$

Graphically we can use the Parallelogram Law:

If you have the vectors in components form you again use:
$\vec{v} - \vec{w} = \vec{v} + \left(- \vec{w}\right)$ operating on each set of corresponding components.

For example:
$\vec{v} = 4 \vec{i} + 2 \vec{j} - 5 \vec{k}$ and:
$\vec{w} = - 2 \vec{i} + 4 \vec{j} + \vec{k}$

$\vec{v} - \vec{w} = \vec{v} + \left(- \vec{w}\right) = \left[4 + \left(2\right)\right] \vec{i} + \left[2 + \left(- 4\right)\right] \vec{j} + \left[- 5 + \left(- 1\right)\right] \vec{k} =$
$= 6 \vec{i} - 2 \vec{j} - 6 \vec{k}$