Question

Let the vectors \(\vec{a}, \vec{b}, \vec{c}\) be such that \(|\vec{a}| = 2\), \(|\vec{b}| = 4\) and \(|\vec{c}| = 4\). If the projection of \(\vec{b}\) on \(\vec{a}\) is equal to the projection of \(\vec{c}\) on \(\vec{a}\) and \(\vec{b}\) is perpendicular to \(\vec{c}\), then the value of \(|\vec{a} + \vec{b} - \vec{c}|\) is ___________.

Hint:

Equal projections give \(\vec{a}\cdot\vec{b} = \vec{a}\cdot\vec{c}\); expand \(|\vec{a} + \vec{b} - \vec{c}|^2\) using dot products.

Answer 1

Correct Answer: 6

Equal projections: \(\dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}|} = \dfrac{\vec{a}\cdot\vec{c}}{|\vec{a}|} \Rightarrow \vec{a}\cdot\vec{b} = \vec{a}\cdot\vec{c}\). Also \(\vec{b}\cdot\vec{c} = 0\).
\(|\vec{a} + \vec{b} - \vec{c}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2\vec{a}\cdot\vec{b} - 2\vec{a}\cdot\vec{c} - 2\vec{b}\cdot\vec{c}\)
\(= 4 + 16 + 16 + 2(\vec{a}\cdot\vec{b} - \vec{a}\cdot\vec{c}) - 0 = 36\).
So \(|\vec{a} + \vec{b} - \vec{c}| = 6\).
Answer: 6