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: \[ \text{Ratio} = 3:2. \] Thus, the correct answer is \( 3:2 \).

Answer 2

Given:
\[ 2\vec{a} + 3\vec{b} + 5\vec{c} - 10\vec{d} = 0 \] Find the ratio in which point \( \vec{c} \) divides the line segment joining \( \vec{a} \) and \( \vec{b} \).

Step 1: Rearrange the equation:
\[ 5 \vec{c} = 10 \vec{d} - 2 \vec{a} - 3 \vec{b} \] \[ \vec{c} = 2 \vec{d} - \frac{2}{5} \vec{a} - \frac{3}{5} \vec{b} \]

Step 2: Assume \( \vec{c} \) divides the line segment \( \vec{a} \vec{b} \) in ratio \( m:n \), so:
\[ \vec{c} = \frac{m \vec{b} + n \vec{a}}{m + n} \] We want to find \( m:n \).

Step 3: Compare with the expression for \( \vec{c} \) in Step 1 ignoring \( \vec{d} \) (since \( \vec{c} \) lies on the line \( \vec{a}\vec{b} \), \( \vec{d} \) must be related accordingly or its contribution zero).

Step 4: Equate coefficients:
\[ \frac{m}{m + n} = -\frac{3}{5}, \quad \frac{n}{m + n} = -\frac{2}{5} \] Since ratios must be positive, multiply numerator and denominator by -1:
\[ \frac{m}{m + n} = \frac{3}{5}, \quad \frac{n}{m + n} = \frac{2}{5} \] So:
\[ \frac{m}{m + n} : \frac{n}{m + n} = \frac{3}{5} : \frac{2}{5} = 3 : 2 \]

Therefore,
\[ \boxed{3 : 2} \]