Question

If $\alpha,\beta,\gamma$ are the angles made by the straight line $\frac{x-5}{2}=\frac{y+1}{3}=\frac{z-2}{\sqrt{7}}$ with the x-axis, y-axis and z-axis respectively, then $\cos\alpha, \cos\beta, \cos\gamma$ are respectively,

• A. $\frac{1}{\sqrt{5}},\frac{7}{2\sqrt{5}},\frac{\sqrt{7}}{2\sqrt{5}}$
• B. $\frac{1}{\sqrt{5}},\frac{3}{2\sqrt{5}},\frac{3\sqrt{7}}{2\sqrt{5}}$
• C. $\frac{-1}{\sqrt{5}},\frac{3}{2\sqrt{5}},\frac{\sqrt{7}}{2\sqrt{5}}$
• D. $\frac{1}{\sqrt{5}},\frac{-3}{2\sqrt{5}},\frac{\sqrt{7}}{2\sqrt{5}}$
• E. $\frac{1}{\sqrt{5}},\frac{3}{2\sqrt{5}},\frac{\sqrt{7}}{2\sqrt{5}}$

Hint:

Math Tip: A quick way to verify direction cosines is to check if the sum of their squares equals 1. Here, $\left(\frac{1}{\sqrt{5}}\right)^2 + \left(\frac{3}{2\sqrt{5}}\right)^2 + \left(\frac{\sqrt{7}}{2\sqrt{5}}\right)^2 = \frac{1}{5} + \frac{9}{20} + \frac{7}{20} = \frac{4+9+7}{20} = 1$. The calculation is correct!

Answer 1

Answer: (E)

Concept:
Three-Dimensional Geometry - Direction Cosines.
If a line has direction ratios $(a, b, c)$, its direction cosines $(l, m, n)$, which correspond to $\cos\alpha, \cos\beta, \cos\gamma$, are given by: $$ l = \frac{a}{\sqrt{a^2+b^2+c^2}}, \quad m = \frac{b}{\sqrt{a^2+b^2+c^2}}, \quad n = \frac{c}{\sqrt{a^2+b^2+c^2}} $$
Step 1: Extract the direction ratios from the line equation.
The given equation is already in the standard symmetric form: $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$.
The denominators represent the direction ratios $(a, b, c)$:

• $a = 2$
• $b = 3$
• $c = \sqrt{7}$

Step 2: Calculate the magnitude of the direction vector.
Find the denominator for the direction cosines formula, $r = \sqrt{a^2 + b^2 + c^2}$: $$ r = \sqrt{2^2 + 3^2 + (\sqrt{7})^2} $$ $$ r = \sqrt{4 + 9 + 7} $$ $$ r = \sqrt{20} $$ Simplify the square root: $$ r = 2\sqrt{5} $$
Step 3: Calculate the direction cosines ($\cos\alpha, \cos\beta, \cos\gamma$).
Divide each direction ratio by the calculated magnitude $r$:

• $\cos\alpha = \frac{a}{r} = \frac{2}{2\sqrt{5}} = \frac{1}{\sqrt{5}}$
• $\cos\beta = \frac{b}{r} = \frac{3}{2\sqrt{5}}$
• $\cos\gamma = \frac{c}{r} = \frac{\sqrt{7}}{2\sqrt{5}}$

Step 4: Format the final answer.
The values of $\cos\alpha, \cos\beta, \cos\gamma$ respectively are: $$ \left( \frac{1}{\sqrt{5}}, \frac{3}{2\sqrt{5}}, \frac{\sqrt{7}}{2\sqrt{5}} \right) $$