Question
If the vectors $$\vec a = \hat i - \hat j + 2\hat k,\,\vec b = 2\hat i + 4\hat j + \,\hat k$$ and $$\vec c = \lambda \hat i + \hat j + \mu \hat k$$ are mutually orthogonal, then $$\left( {\lambda ,\,\mu } \right) = ?$$
A.
$$\left( {2,\, - 3} \right)$$
B.
$$\left( { - 2,\, 3} \right)$$
C.
$$\left( {3,\, - 2} \right)$$
D.
$$\left( { - 3,\, 2} \right)$$
Answer :
$$\left( { - 3,\, 2} \right)$$
Solution :
Since, $$\vec a,\,\vec b$$ and $${\vec c}$$ are mutually orthogonal
$$\eqalign{
& \therefore \,\,\vec a.\vec b = 0,\,\,\vec b.\vec c = 0,\,\,\vec c.\vec a = 0 \cr
& \Rightarrow 2\lambda + 4 + \mu = 0.....({\text{i}}) \cr
& \Rightarrow \lambda - 1 + 2\mu = 0.....({\text{ii}}) \cr} $$
On solving (i) and (ii), we get $$\lambda = - 3,\,\,\mu = 2$$