If \(\vec{a}\) and \(\vec{b}\) are two perpendicular vectors such that \(|\vec{a}| = 3\), \(|\vec{b}| = 4\) and θ is the angle between \(\vec{a}\) and \((\vec{a} - \vec{b})\), then cos θ is equal to:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{3}{5}\)
Since \(\vec{a}\) and \(\vec{b}\) are perpendicular, \(\vec{a}\cdot\vec{b} = 0\). Then
\(\vec{a}\cdot(\vec{a} - \vec{b}) = |\vec{a}|^2 - \vec{a}\cdot\vec{b} = 9 - 0 = 9\)
\(|\vec{a} - \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 - 2\,\vec{a}\cdot\vec{b} = 9 + 16 = 25\)
\(|\vec{a} - \vec{b}| = 5\)
By the dot product formula for the angle:
\( \displaystyle \cos\theta = \frac{\vec{a}\cdot(\vec{a} - \vec{b})}{|\vec{a}|\,|\vec{a} - \vec{b}|} = \frac{9}{3 \times 5} = \frac{3}{5} \)