Question

The projection of a directed line segment on the co-ordinate axes are $12,4,3$, the DC's of the line are

• A. $\frac{-12}{13},\frac{-4}{13},\frac{-3}{13}$
• B. $\frac{12}{13},\frac{4}{13},\frac{3}{13}$
• C. $\frac{12}{13},\frac{-4}{13},\frac{3}{13}$
• D. $\frac{12}{13},\frac{4}{13},\frac{-3}{13}$

Answer 1

The correct option is B $\frac{12}{13},\frac{4}{13},\frac{3}{13}$

Let the line segment be represented in vector form as $\overrightarrow{a}=a_1\overrightarrow{i}+a_2\overrightarrow{j}+a_3\overrightarrow{k}$

Vector along coordinate axes are $\overrightarrow{i},\overrightarrow{j},\overrightarrow{k}$

Given that projection of $\overrightarrow{a}$ on x axis is 12, that with y axis is 4 and that with z axis is 3.

$⟹\overrightarrow{a}.\overrightarrow{i}=12,\therefore a_1=12$

similarily, $a_2=4,a_3=3$

$\therefore \overrightarrow{a}=12\overrightarrow{i}+4\overrightarrow{j}+3\overrightarrow{k}$

the length of the line segment=$|\overrightarrow{a}|=\sqrt{144+16+9}=13$

$\therefore |PQ|=13$

Thus the direction cosines of PQ are $\frac{12}{13},\frac{4}{13},\frac{3}{13}$