Question

If the straight line passing through the points $(1,-1,2)$ and $(3,2,8)$ makes angle $\beta$ with the $y$-axis, then the value of $\cos \beta$ is equal to

• A. $\frac{1}{7}$
• B. $\frac{3}{7}$
• C. $\frac{2}{7}$
• D. $\frac{4}{7}$
• E. $\frac{5}{7}$

Hint:

The triplet (2, 3, 6) is very common in 3D problems because $2^2 + 3^2 + 6^2 = 49$, which is a perfect square ($7^2$). Memorizing this can speed up your calculations.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The angle a line makes with the $y$-axis is given by its second direction cosine ($m = \cos \beta$). Direction cosines are found by normalizing the direction ratios of the line.
Key Formula or Approach:
1. Direction Ratios (D.R.s): $(x_2 - x_1, y_2 - y_1, z_2 - z_1)$.
2. Direction Cosines (D.C.s): $\frac{b}{\sqrt{a^2 + b^2 + c^2}}$ for the $y$-axis component.

Step 2: Detailed Explanation:
Given points: $A(1, -1, 2)$ and $B(3, 2, 8)$.
Direction Ratios of line $AB$:
$a = 3 - 1 = 2$
$b = 2 - (-1) = 3$
$c = 8 - 2 = 6$
Magnitude of the direction vector:
\[ L = \sqrt{a^2 + b^2 + c^2} = \sqrt{2^2 + 3^2 + 6^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7 \]
The direction cosine $m$ with respect to the $y$-axis is:
\[ \cos \beta = \frac{b}{L} = \frac{3}{7} \]

Step 3: Final Answer:
The value of $\cos \beta$ is $3/7$.