Question

If the position vector of a point A is \( \vec{A} = \vec{a} + 2\vec{b} \) and \( \vec{a} \) divides \( AB \) in the ratio 2 : 3, then the position vector of B is

• A. \( \vec{b} \)
• B. \( 2\vec{a} - \vec{b} \)
• C. \( \vec{b} - 2\vec{a} \)
• D. \( \vec{a} - 3\vec{b} \)

Hint:

When solving problems involving the position vectors of points and ratios, always use the section formula to divide the line segment in the given ratio. This allows you to find the position vector of any point on the line.

Answer 1

The Correct Option is D

We are given the following: - The position vector of point \( A \) is \( \vec{A} = \vec{a} + 2\vec{b} \). - The point \( A \) divides the line segment \( AB \) in the ratio 2 : 3. We need to find the position vector of point \( B \). Step 1: Use the section formula The section formula for the position vector of a point dividing a line segment \( AB \) in the ratio \( m : n \) is: \[ \vec{P} = \frac{n \vec{A} + m \vec{B}}{m + n} \] Here, point \( A \) divides \( AB \) in the ratio 2 : 3, so: \[ \vec{A} = \frac{3 \vec{B} + 2 \vec{A}}{2 + 3} \] Simplifying this formula, we get: \[ \vec{A} = \frac{3 \vec{B} + 2 \vec{A}}{5} \] Now, multiply through by 5: \[ 5 \vec{A} = 3 \vec{B} + 2 \vec{A} \] Now, subtract \( 2 \vec{A} \) from both sides: \[ 5 \vec{A} - 2 \vec{A} = 3 \vec{B} \] \[ 3 \vec{A} = 3 \vec{B} \] Finally, divide both sides by 3: \[ \vec{A} = \vec{B} \] Thus the correct answer is \( \boxed{4} \).