Question

A line segment \( PQ \) has the length 63 and direction ratios \( (3, -2, 6) \). If this line makes an obtuse angle with the X-axis, then the components of the vector \( \vec{PQ} \) are:

• A. \( 7, 8, -4 \)
• B. \( -7, 8, -4 \)
• C. \( 27, -18, 54 \)
• D. \( -27, 18, -54 \)

Hint:

For direction ratio problems, normalize the vector using the magnitude constraint, and check for obtuse angles by ensuring the dot product with the X-axis is negative.

Answer 1

The Correct Option is D

The vector \( \vec{PQ} \) is given by: \[ \vec{PQ} = \lambda (3, -2, 6) \] where \( \lambda \) is found using the length constraint: \[ \sqrt{(3\lambda)^2 + (-2\lambda)^2 + (6\lambda)^2} = 63 \] Solving for \( \lambda \), then ensuring the vector makes an obtuse angle with the X-axis, \[ \vec{PQ} = (-27, 18, -54) \]