Question

If vectors a, b, c are three mutually perpendicular vectors of equal magnitude, prove that vectors a + b + c is equally inclined with vectors a, b and c.

Answer 1

\(\vec a . \vec b = \vec b.\vec c = \vec c . \vec a = 0\)

Let the angle between \(\vec a\) and \((\vec a + \vec b + \vec c) \) be α

\(\vec a .(\vec a + \vec b + \vec c) = |\vec a||\vec a + \vec b + \vec c| \cos \alpha\)

\(\vec a . \vec a + \vec a. \vec b + \vec a.\vec c= |\vec a||\vec a + \vec b + \vec c| \cos \alpha\)

\(|\vec a|^2 + 0 + 0= |\vec a||\vec a + \vec b + \vec c| \cos \alpha\)

⇒ \(\cos \alpha = \frac{|\vec a|}{|\vec a + \vec b + \vec c|}\)

Similarly angle between \(\vec b\) and \((\vec a + \vec b + \vec c)\) be \(\beta\)

⇒ \(\cos \beta = \frac{|\vec b|}{|\vec a + \vec b + \vec c|}\)

And angle between \(\vec c\) and \((\vec a + \vec b + \vec c)\) be \(\gamma\)

⇒ \(\cos \gamma = \frac{|\vec c|}{|\vec a + \vec b + \vec c|}\)

As all the vectors are equal in magnitude

⇒ \(\cos \alpha = \cos \beta =\cos \gamma\)

⇒ \(\alpha = \beta = \gamma\)

Hence proved.

Answer 2

Hence,  is equally inclineded with vector a, b and c.