Question

Find the values of x and y if the vectors \[\vec{a} = 3 \hat{i} + x \hat{j} - \hat{k} \text{ and } \vec{b} = 2 \hat{i} + \hat{j} + y \hat{k}\] are mutually perpendicular vectors of equal magnitude. 

Answer 1

\[\text{ We have }\]

\[ \vec{a} = 3 \hat{i} + x \hat{j} - \hat{k} \text{ and } \vec{b} = 2 \hat{i} + \hat{j} + y \hat{k} \]

\[\text{It is given that the vectors are perpendicular}.\]

\[ \Rightarrow \vec{a} . \vec{b} = 0\]

\[ \Rightarrow 6 + x - y = 0\]

\[ \Rightarrow x - y = - 6 . . . \left( 1 \right)\]

\[\text{ Also, it is given that }\]

\[\left| \vec{a} \right| = \left| \vec{b} \right|\]

\[ \Rightarrow \sqrt{9 + x^2 + 1} = \sqrt{4 + 1 + y^2}\]

\[ \Rightarrow \sqrt{10 + x^2} = \sqrt{5 + y^2}\]

\[ \Rightarrow 10 + x^2 = 5 + y^2 \]

\[ \Rightarrow x^2 - y^2 = - 5\]

\[ \Rightarrow \left( x + y \right)\left( x - y \right) = - 5\]

\[ \Rightarrow - 6 \left( x + y \right) = - 5 .........................\left[\text{ Using } \left( 1 \right) \right]\]

\[ \Rightarrow x + y = \frac{5}{6} . . . \left( 2 \right)\]

\[(1)+(2) \text{ gives }\]

\[2x = \frac{- 31}{6}\]

\[ \Rightarrow x = \frac{- 31}{12}\]

\[\text{ From } (1),\]

\[\frac{- 31}{12} - y = - 6\]

\[ \Rightarrow y = \frac{- 31}{12} + 6 = \frac{41}{12}\]

\[ \therefore x = \frac{- 31}{12} \text{ and } y = \frac{41}{12}\]