Question

Show that the sum of the length of projections of ${\displaystyle \text{p}\hat{\text{i}}+\text{q}\hat{\text{j}}+\text{r}\hat{\text{k}}}$ on the coordinate axes, where p = 2, q = 3 and r = 4 is 9.

Answer 1

Let ${\displaystyle \overline{\text{a}}=\text{p}\hat{\text{i}}+\text{q}\hat{\text{j}}+\text{r}\hat{\text{k}}}$

Projection of ${\displaystyle \overline{\text{a}}}$ on X-axis

${\displaystyle =\frac{\overline{\text{a}}.\hat{\text{i}}}{\left|\hat{\text{i}}\right|}=\frac{(\text{p}\hat{\text{i}}+\text{q}\hat{\text{i}}+\text{r}\hat{\text{k}}).\hat{\text{i}}}{1}}$ = p = 2

Similarly, projections of ${\displaystyle \overline{\text{a}}}$ on Y- and Z-axes are 3 and 4 respectively.

∴ sum of these projections = 2 + 3 + 4 = 9.