Question

The projection of the line segment joining \( (2,0,-3) \) and \( (5,-1,2) \) on a straight line whose direction ratios are \( 2,4,4 \) is equal to

• A. \( \frac{11}{6} \)
• B. \( \frac{10}{3} \)
• C. \( \frac{13}{3} \)
• D. \( \frac{13}{6} \)
• E. \( \frac{11}{3} \)

Hint:

Projection = component of one vector along another, not the vector itself.

Answer 1

Answer: (E)

Concept:
• Projection of a vector \( \vec{a} \) on a direction \( \vec{b} \) is: \[ \text{Projection} = \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|} \]
• Here, the line segment gives a vector, and direction ratios give direction vector.

Step 1: Convert points into vector form.
Let: \[ A = (2,0,-3), \quad B = (5,-1,2) \] The vector representing the line segment is: \[ \vec{AB} = \vec{OB} - \vec{OA} \] \[ = (5-2,\,-1-0,\;2-(-3)) \] \[ = (3,\,-1,\;5) \]

Step 2: Interpret direction ratios.
Given direction ratios: \[ 2,4,4 \] So direction vector: \[ \vec{d} = (2,4,4) \]

Step 3: Understand projection physically.
Projection means:
• Shadow (component) of vector \( \vec{AB} \) along direction \( \vec{d} \)
• Only the component in that direction is considered

Step 4: Compute dot product.
\[ \vec{AB} \cdot \vec{d} = 3\cdot2 + (-1)\cdot4 + 5\cdot4 \] \[ = 6 - 4 + 20 = 22 \]

Step 5: Compute magnitude of direction vector.
\[ |\vec{d}| = \sqrt{2^2 + 4^2 + 4^2} \] \[ = \sqrt{4 + 16 + 16} = \sqrt{36} = 6 \]

Step 6: Apply projection formula.
\[ \text{Projection} = \frac{22}{6} \] \[ = \frac{11}{3} \]

Step 7: Interpret result.
This represents:
• Scalar projection (length of shadow)
• Always along direction vector

Step 8: Final Answer.
\[ \boxed{\frac{11}{3}} \]