Unit Vectors and Vector Addition — Analytical Method

A unit vector along any direction is a vector of magnitude exactly 1, pointing in that

direction, and carrying no physical unit of its own — it exists purely to specify "which

way." The unit vectors along the positive x-axis and positive y-axis are given the standard

symbols $\hat i$ and $\hat j$ respectively (a third unit vector $\hat k$, along the z-axis,

completes the set in three dimensions but is not needed for motion confined to a plane).

Using these standard unit vectors, any vector $\vec A$ lying in the plane can be written

in terms of its rectangular components (Section 3.6) as

This way of writing a vector is called its component form, and it is what makes the

analytical (component) method of vector addition possible: instead of drawing triangles

or parallelograms to scale, two vectors are added simply by adding their corresponding

components,

and subtracted by subtracting components,

Multiplying a vector by a scalar (Section 3.4) likewise multiplies each component:

$\lambda\vec A = \lambda A_x\hat i + \lambda A_y\hat j$.

The magnitude and direction of any combination can then be recovered at the end from its

final components, exactly as in Section 3.6: …