23. \( A(1,2) \) and \( B(m,-2) \) are the terminal points of the vector \( \overrightarrow{A B} \). If \( |\overrightarrow{A B}|=5 \) then nd find the value of \( m \).

To find the value of \( m \), we can use the formula for the distance between two points in a coordinate plane. The distance \( |\overrightarrow{AB}| \) between points \( A(1, 2) \) and \( B(m, -2) \) is given by: \[ |\overrightarrow{A B}| = \sqrt{(m - 1)^2 + (-2 - 2)^2} \] Given that \( |\overrightarrow{A B}| = 5 \), we can set up the equation: \[ \sqrt{(m - 1)^2 + (-4)^2} = 5 \] This simplifies to: \[ \sqrt{(m - 1)^2 + 16} = 5 \] Squaring both sides, we have: \[ (m - 1)^2 + 16 = 25 \] Subtracting 16 from both sides gives: \[ (m - 1)^2 = 9 \] Taking the square root of both sides, we get two possible equations: \[ m - 1 = 3 \quad \text{or} \quad m - 1 = -3 \] Solving these, we find: 1. \( m - 1 = 3 \) → \( m = 4 \) 2. \( m - 1 = -3 \) → \( m = -2 \) Thus, the possible values for \( m \) are \( m = 4 \) or \( m = -2 \).