Question

The projection of the line segment joining P(2, -1, 0) and Q(3, 2, -1) on the line whose direction ratios are 1, 2, 2 is

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

Hint:

Projection of $\vec{A}$ on $\vec{B} = \frac{|\vec{A} \cdot \vec{B}|}{|\vec{B}|}$.

Answer 1

The Correct Option is D

Step 1: Vector PQ
$\vec{PQ} = (3-2, 2+1, -1-0) = (1, 3, -1)$.
Step 2: Direction Cosines of Line
$DRs = (1, 2, 2)$. Magnitude $= \sqrt{1^2+2^2+2^2} = 3$.
$DCs = (1/3, 2/3, 2/3)$.
Step 3: Projection
Projection $= |x \cdot l + y \cdot m + z \cdot n| = |1(1/3) + 3(2/3) + (-1)(2/3)|$.
$= |1/3 + 6/3 - 2/3| = 5/3$.
Final Answer: (D)