Question

The straight line passing through the points $(3,2,3)$ and $(5,-1,-2)$ is perpendicular to the straight line passing through the points $(1,3,1)$ and $(\alpha, \alpha, \alpha)$. Then the value of $\alpha$ is equal to

• A. 1
• B. 2
• C. 3
• D. 4
• E. 5

Hint:

In 3D geometry, "perpendicular" always translates to "Dot Product = 0". Use this immediately after finding direction ratios.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Two lines are perpendicular if the dot product of their direction vectors is zero ($a_1 a_2 + b_1 b_2 + c_1 c_2 = 0$).

Step 2: Detailed Explanation:
First line passes through $A(3, 2, 3)$ and $B(5, -1, -2)$.
Direction vector $\vec{d_1} = (5-3, -1-2, -2-3) = (2, -3, -5)$.
Second line passes through $C(1, 3, 1)$ and $D(\alpha, \alpha, \alpha)$.
Direction vector $\vec{d_2} = (\alpha-1, \alpha-3, \alpha-1)$.
Condition for perpendicularity: $\vec{d_1} \cdot \vec{d_2} = 0$.
\[ 2(\alpha - 1) + (-3)(\alpha - 3) + (-5)(\alpha - 1) = 0 \]
Expand the terms:
\[ 2\alpha - 2 - 3\alpha + 9 - 5\alpha + 5 = 0 \]
Combine like terms:
\[ (2 - 3 - 5)\alpha + (-2 + 9 + 5) = 0 \]
\[ -6\alpha + 12 = 0 \]
\[ 6\alpha = 12 \]
\[ \alpha = 2 \]

Step 3: Final Answer:
The value of $\alpha$ is 2.