Question

If \(\vec a, \vec b, \vec c, \vec d\) are the position vectors of the points \(A,B,C,D\) respectively such that \[ 3\vec a - \vec b + 2\vec c - 4\vec d = \vec 0, \] then the position vector of the point of intersection of the line segments \(AC\) and \(BD\) is

• A. \( \dfrac{\vec b + 3\vec d}{4} \)
• B. \( \dfrac{3\vec a + \vec c}{4} \)
• C. \( \dfrac{\vec a + \vec c}{2} \)
• D. \( \dfrac{\vec b + 4\vec d}{5} \)

Hint:

Vector equations of the form \(m\vec a + n\vec c = p\vec b + q\vec d\) often indicate intersection points.

Answer 1

The Correct Option is D

Step 1: Use the given vector relation.
\[ 3\vec a + 2\vec c = \vec b + 4\vec d \]
Step 2: Interpret the equation.
This equation represents a point dividing both segments \(AC\) and \(BD\) internally in the same ratio.

Step 3: Find the position vector of the intersection point.
\[ \vec r = \frac{\vec b + 4\vec d}{5} \]