Unit Vector

A unit vector is a vector whose length is 1. For instance, i, j, and k are all unit vectors. In general, if \(\mathbf{a} \ne \mathbf{0}\), then the unit vector that has the same direction as a is \[ \mathbf{u} = \frac{1}{|\mathbf{a}|}\mathbf{a} = \frac{\mathbf{a}}{|\mathbf{a}|} \tag{4} \] In order to verify this, we let \(c = 1/|\mathbf{a}|\). Then \(\mathbf{u} = c\mathbf{a}\) and \(c\) is a positive scalar, so u has the same direction as a. Also \[ |\mathbf{u}| = |c\mathbf{a}| = |c||\mathbf{a}| = \frac{1}{|\mathbf{a}|}|\mathbf{a}| = 1 \]

EXAMPLE 6 Find the unit vector in the direction of the vector \(2\mathbf{i} - \mathbf{j} - 2\mathbf{k}\).