Question

If \( \overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} \) are the position vectors of the points A, B, C respectively and \( 5\overrightarrow{a} + 3\overrightarrow{b} - 8\overrightarrow{c} = 0 \), then find the ratio in which the point C divides the line segment AB.

Hint:

The point dividing the line segment in the ratio \( m:n \) has the position vector \( \frac{n\overrightarrow{a} + m\overrightarrow{b}}{m+n} \).

Answer 1

We are given the vector equation: \[ 5\overrightarrow{a} + 3\overrightarrow{b} - 8\overrightarrow{c} = 0 \] Rearranging the equation: \[ 5\overrightarrow{a} + 3\overrightarrow{b} = 8\overrightarrow{c} \] This can be interpreted as the vector form of the point \( C \) dividing the line segment joining \( A \) and \( B \). The formula for the position vector of the point dividing a line segment in the ratio \( m:n \) is: \[ \overrightarrow{c} = \frac{n\overrightarrow{a} + m\overrightarrow{b}}{m+n} \] Comparing this with our equation, we get: \[ 5\overrightarrow{a} + 3\overrightarrow{b} = 8\overrightarrow{c} \] Thus, the ratio in which \( C \) divides \( AB \) is \( 5:3 \). So, the ratio is: \[ \boxed{5:3} \]