Question

If \( \vec{a}, \vec{b}, \vec{c}, \vec{d} \) are position vectors of 4 points such that \( 2\vec{a} + 3\vec{b} + 5\vec{c} - 10\vec{d} = 0 \), then the ratio in which the line joining \( \vec{c} \) divides the line segment joining \( \vec{a} \) and \( \vec{b} \) is:

• A. \( 2:3 \)
• B. \( -1:2 \)
• C. \( 2:1 \)
• D. \( 3:2 \)

Hint:

For problems involving division of a line segment by a point, express the point as a weighted average of the two endpoints, and use the given conditions to find the ratio.

Answer 1

The Correct Option is D

We are given the vector equation: \[ 2\vec{a} + 3\vec{b} + 5\vec{c} - 10\vec{d} = 0. \] Rearrange the equation to solve for \( \vec{c} \): \[ 5\vec{c} = 10\vec{d} - 2\vec{a} - 3\vec{b}. \] Thus, we can express \( \vec{c} \) as a linear combination of \( \vec{a} \), \( \vec{b} \), and \( \vec{d} \), and the ratio in which the line joining \( \vec{c} \) divides the segment joining \( \vec{a} \) and \( \vec{b} \) can be found as: \[ {Ratio} = 3:2. \] Thus, the correct answer is \( 3:2 \).