Question

The projection of the line segment joining the points (–1, 0, 3) and (2, 5, 1) on the line whose direction ratios are 6, 2, 3 is –

• A. $\frac{22}{7}$
• B. $\frac{23}{7}$
• C. $\frac{24}{7}$
• D. $\frac{25}{7}$

Answer 1

Answer: (A)

$\frac{22}{7}$

Answer 2

The direction cosines l, m, n of the line are given by

$\frac{l}{6}$ = $\frac{m}{2}$ = $\frac{n}{3}$ = $\frac{\sqrt{l^{2} + m^{2} + n^{2}}}{\sqrt{6^{2} + 2^{2} + 3^{2}}}$=$\frac{1}{\sqrt{49}}$ = $\frac{1}{7}$

\ l = $\frac{6}{7}$, m = $\frac{2}{7}$, n = $\frac{3}{7}$

The required projection is given by

= l (x₂ – x₁) + m(y₂ – y₁) + n(z₂ – z₁)

= $\frac{6}{7}$ [2 – (–1)] + $\frac{2}{7}$ (5 – 0) + $\frac{3}{7}$ (1 – 3)

= $\frac{6}{7}$× 3 + $\frac{2}{7}$× 5 + $\frac{3}{7}$× (–2)

= $\frac{18}{7}$+ $\frac{10}{7}$ – $\frac{6}{7}$ = $\frac{18 + 10 - 6}{7}$ = $\frac{22}{7}$.

Hence (1) is the correct answer.