Question

Find the unit vector in the direction of
a. −3i + 7j
b. 8i − j + 4k

Answer 1

To find the unit vector in the direction of a given vector \(\mathbf{v}\), we first need to compute the magnitude of \(\mathbf{v}\) and then divide \(\mathbf{v}\) by its magnitude. The magnitude of a vector \(\mathbf{v} = ai + bj + ck\) is given by the formula:

\[
|\mathbf{v}| = \sqrt{a^2 + b^2 + c^2}
\]

After finding the magnitude, the unit vector \(\mathbf{u}\) in the direction of \(\mathbf{v}\) is calculated as:

\[
\mathbf{u} = \frac{\mathbf{v}}{|\mathbf{v}|}
\]

### a. For the vector \(-3i + 7j\):

1. **Identify the components**:
\[
\mathbf{v} = -3i + 7j \quad \Rightarrow \quad a = -3, \, b = 7, \, c = 0
\]

2. **Calculate the magnitude**:
\[
|\mathbf{v}| = \sqrt{(-3)^2 + 7^2 + 0^2} = \sqrt{9 + 49} = \sqrt{58}
\]

3. **Find the unit vector**:
\[
\mathbf{u} = \frac{-3i + 7j}{\sqrt{58}} = -\frac{3}{\sqrt{58}}i + \frac{7}{\sqrt{58}}j
\]

Thus, the unit vector in the direction of \(-3i + 7j\) is:
\[
\mathbf{u} = -\frac{3}{\sqrt{58}}i + \frac{7}{\sqrt{58}}j
\]

### b. For the vector \(8i - j + 4k\):

1. **Identify the components**:
\[
\mathbf{v} = 8i - j + 4k \quad \Rightarrow \quad a = 8, \, b = -1, \, c = 4
\]

2. **Calculate the magnitude**:
\[
|\mathbf{v}| = \sqrt{8^2 + (-1)^2 + 4^2} = \sqrt{64 + 1 + 16} = \sqrt{81} = 9
\]

3. **Find the unit vector**:
\[
\mathbf{u} = \frac{8i - j + 4k}{9} = \frac{8}{9}i - \frac{1}{9}j + \frac{4}{9}k
\]

Thus, the unit vector in the direction of \(8i - j + 4k\) is:
\[
\mathbf{u} = \frac{8}{9}i - \frac{1}{9}j + \frac{4}{9}k
\]

In summary:
- a. The unit vector for \(-3i + 7j\) is \(-\frac{3}{\sqrt{58}}i + \frac{7}{\sqrt{58}}j\).
- b. The unit vector for \(8i - j + 4k\) is \(\frac{8}{9}i - \frac{1}{9}j + \frac{4}{9}k\).