Vectors: Question 1

Worked solution

Step 1: Find the magnitude of $\mathbf{p}$

$|\mathbf{p}| = \sqrt{4^2 + (-1)^2 + 8^2} = \sqrt{16 + 1 + 64} = \sqrt{81} = 9$

Step 2: Divide each component by the magnitude

The unit vector $\hat{\mathbf{p}}$ in the direction of $\mathbf{p}$ is $\mathbf{p}$ scaled by $\dfrac{1}{|\mathbf{p}|}$:

$\hat{\mathbf{p}} = \frac{\mathbf{p}}{|\mathbf{p}|} = \frac{4}{9}\mathbf{i} - \frac{1}{9}\mathbf{j} + \frac{8}{9}\mathbf{k}$

Step 3: Recompute independently as a check

Squaring and summing the components in a different order: $8^2 + 4^2 + (-1)^2 = 64 + 16 + 1 = 81$, so $|\mathbf{p}| = \sqrt{81} = 9$, the same result as Step 1.

As a further check, the magnitude of $\hat{\mathbf{p}}$ itself must equal $1$:

$\left(\frac{4}{9}\right)^2 + \left(\frac{1}{9}\right)^2 + \left(\frac{8}{9}\right)^2 = \frac{16 + 1 + 64}{81} = \frac{81}{81} = 1 \checkmark$

Why the other options are wrong

• B: comes from dividing by $|\mathbf{p}|^2 = 81$ instead of $|\mathbf{p}| = 9$. The resulting vector has magnitude $\dfrac{1}{9}$, not $1$.
• C: comes from treating $4 + (-1) + 8 = 11$ as the “magnitude”, skipping the squaring and square-rooting entirely.
• D: has the correct size ($\dfrac{4}{9}$, $\dfrac{1}{9}$, $\dfrac{8}{9}$) but every sign flipped, so it points in the opposite direction to $\mathbf{p}$.

Final answer

$\boxed{\hat{\mathbf{p}} = \frac{4}{9}\mathbf{i} - \frac{1}{9}\mathbf{j} + \frac{8}{9}\mathbf{k}} \quad \text{(Option A)}$