Question

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

• A. \( \frac{19}{7} \) units
• B. \( \frac{17}{7} \) units
• C. \( \frac{11}{7} \) units
• D. \( \frac{15}{7} \) units

Hint:

The projection of a segment on a line is the dot product of the segment vector with the unit vector along the line.

Answer 1

The Correct Option is A

Step 1: Unit vector
The unit vector along the line is $\frac{1}{7}(3,2,6)$.

Step 2: Dot product
The segment vector is $(-3,-1,5)$, so the projection is $\frac{1}{7}(-9-2+30)$.

Step 3: Result
This equals $\frac{19}{7}$ units, matching the first option.

Final Answer:
The required projection is $\frac{19}{7}$ units. \[ \boxed{\text{(A) }\dfrac{19}{7}\text{ units}} \]