6 The magnitude of \( \overrightarrow{\mathrm{PQ}} \) is 20 units. Given that \( \overrightarrow{\mathrm{OP}}=\binom{5}{0} \) and \( \overrightarrow{\mathrm{OQ}}=\binom{17}{y} \), find the positive value of \( y \).

To find the positive value of \( y \), we can use the magnitude formula for vectors. The magnitude of the vector \( \overrightarrow{\mathrm{PQ}} \) can be computed using the points \( \overrightarrow{\mathrm{OP}} \) and \( \overrightarrow{\mathrm{OQ}} \) as follows: 1. First, express the vector \( \overrightarrow{\mathrm{PQ}} \): \[ \overrightarrow{\mathrm{PQ}} = \overrightarrow{\mathrm{OQ}} - \overrightarrow{\mathrm{OP}} = \binom{17}{y} - \binom{5}{0} = \binom{17 - 5}{y - 0} = \binom{12}{y} \] 2. The magnitude of \( \overrightarrow{\mathrm{PQ}} \) is given by: \[ |\overrightarrow{\mathrm{PQ}}| = \sqrt{(12)^2 + (y - 0)^2} = \sqrt{144 + y^2} \] 3. Setting this equal to the given magnitude of 20 units: \[ \sqrt{144 + y^2} = 20 \] 4. Squaring both sides: \[ 144 + y^2 = 400 \] 5. Solving for \( y^2 \): \[ y^2 = 400 - 144 = 256 \] 6. Taking the positive square root: \[ y = \sqrt{256} = 16 \] Thus, the positive value of \( y \) is \( 16 \).