If vector(a), vector(b), vector(c) are mutually perpendicular vectors of equal magnitudes, show that the vector

Question

If \(\vec{a},\vec{b},\vec{c}\) are mutually perpendicular vectors of equal magnitudes, show that the vector \(\vec{a}+\vec{b}+\vec{c}\) is equally inclined to \(\vec{a},\vec{b}\) and \(\vec{c}.\) Also find the angle which \(\vec{a}+\vec{b}+\vec{c}\) makes with \(\vec{a}\) or \(\vec{b}\) or \(\vec{c}\)

Answer 1

Since \(\vec{a},\vec{b},\vec{c}\) are mutually perpendicular vectors. Therefore,

Let \(\theta_1\) and \(\theta_2\) and \(\theta_3\) be the angles made by \((\vec{a}+\vec{b}+\vec{c})\) with \(\vec{a},\vec{b}\) and \(\vec{c}\) respectively.

Similarly, we have \(\theta_2=cos^{-1}(\frac{1}{\sqrt{3}})\) and \(\theta_3=cos^{-1}(\frac{1}{\sqrt{3}})\)

\(i.e.,\)

\((\vec{a}+\vec{b}+\vec{c})\) is equally inclined with \(\vec{a},\vec{b}\) and \(\vec{c}\)