Question:
The length of the perpendicular from the point \( (1, 2, 3) \) on the line \( \frac{x - 6}{3} = \frac{y - 7}{2} = \frac{z - 7}{-2} \) is :
The length of the perpendicular from the point \( (1, 2, 3) \) on the line \( \frac{x - 6}{3} = \frac{y - 7}{2} = \frac{z - 7}{-2} \) is :
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Distance of a point from a line = |MP x d| / |d| where M is a point on the line and d its direction vector.
Updated On: Nov 12, 2026
- 5
- 6
- 4
- 7
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The Correct Option is D
Solution and Explanation
A point on the line is \( M(6, 7, 7) \) and the direction vector is \( \vec{d} = (3, 2, -2) \), \( |\vec{d}| = \sqrt{17} \).
\( \overrightarrow{MP} = (1 - 6, 2 - 7, 3 - 7) = (-5, -5, -4) \).
\( \overrightarrow{MP} \times \vec{d} = (10 + 8, -(10 + 12), -10 + 15) = (18, -22, 5) \), with squared magnitude \( 324 + 484 + 25 = 833 \).
Distance = \( \frac{\sqrt{833}}{\sqrt{17}} = \sqrt{49} = 7 \).
Answer: (4) 7
\( \overrightarrow{MP} = (1 - 6, 2 - 7, 3 - 7) = (-5, -5, -4) \).
\( \overrightarrow{MP} \times \vec{d} = (10 + 8, -(10 + 12), -10 + 15) = (18, -22, 5) \), with squared magnitude \( 324 + 484 + 25 = 833 \).
Distance = \( \frac{\sqrt{833}}{\sqrt{17}} = \sqrt{49} = 7 \).
Answer: (4) 7
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