Question

If \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are the position vectors of the points \( A, B, C \) respectively and \[ 5\mathbf{a} - 3\mathbf{b} - 2\mathbf{c} = \mathbf{0}, \] then find the ratio in which the point \( C \) divides the line segment \( BA \) externally.

Hint:

For a point dividing a line segment \( BA \) externally in the ratio \( m:n \), the position vector is given by: \[ \mathbf{c} = \frac{m\mathbf{a} - n\mathbf{b}}{m - n}. \]

Answer 1

Step 1: Define the Position Vector of \( C \)
Let \( C \) divide \( BA \) externally in the ratio \( 5:3 \). The position vector of \( C \) can be expressed as: \[ \mathbf{c} = \frac{5\mathbf{a} - 3\mathbf{b}}{5 - 3}. \] Step 2: Use the Given Equation
Rearranging the given equation \( 5\mathbf{a} - 3\mathbf{b} - 2\mathbf{c} = \mathbf{0} \), we obtain: \[ 2\mathbf{c} = 5\mathbf{a} - 3\mathbf{b}. \] Step 3: Substitute the Expression for \( \mathbf{c} \)
Substitute the previously derived expression for \( \mathbf{c} \): \[ 2 \times \frac{5\mathbf{a} - 3\mathbf{b}}{5 - 3} = 5\mathbf{a} - 3\mathbf{b}. \] Simplifying the expression gives: \[ \mathbf{c} = \frac{5\mathbf{a} - 3\mathbf{b}}{2}. \] Thus, point \( C \) divides the line segment \( BA \) externally in the ratio \( 5:3 \).