Equality of Vectors

Two vectors are said to be equal if, and only if, they have the same magnitude and point

in exactly the same direction. Where the vector is actually drawn in the plane does not

matter for this comparison — a vector is a "free" quantity that can be shifted parallel to

itself without changing its identity (position vectors are a special, "bound" exception,

since they are always measured from a fixed origin, as noted in the previous section).

In component form, $\vec A = A_x\hat i + A_y\hat j$ and $\vec B = B_x\hat i + B_y\hat j$ are

equal if and only if their corresponding components are equal:

This is often the quickest way to check equality without needing to compute magnitudes and

angles separately, though computing the magnitude ($A = \sqrt{A_x^2+A_y^2}$) and the

direction angle ($\theta = \tan^{-1}(A_y/A_x)$) and comparing those between the two vectors

works equally well.

Two vectors that have the same magnitude but opposite directions are not equal to each

other — each is called the negative of the other. If $\vec B = -\vec A$, then $\vec B$ …