Question

If \[\vec a = 2\hat i + \lambda \hat j - 3\hat k\] and \[\vec b = 4\hat i - 3\hat j - 2\hat k\] are perpendicular to each other then find the value of scalar \[\lambda \].

Answer 1

Hint: When two vectors are perpendicular to each other then the dot product between those vectors will be equal to zero i.e. \[\vec a\]∙\[\vec b\]=0.

Complete step by step answer:
Given \[\vec a = 2\hat i + \lambda \hat j - 3\hat k\]
           \[\vec b = 4\hat i - 3\hat j - 2\hat k\]
Now according to question,
Both the vectors \[\vec a\] and \[\vec b\] are perpendicular to each other.
∴ Their dot product = 0
Now we will multiply the two vectors according to dot product rules
So we have, \[\vec a\]∙\[\vec b\]=0
\[
   \Rightarrow (2\hat i + \lambda \hat j - 3\hat k).(4\hat i - 3\hat j - 2\hat k) = 0 \\
   \Rightarrow 2\hat i.4\hat i + \lambda \hat j.\left( { - 3\hat j} \right) + \left( { - 3\hat k} \right).\left( { - 2\hat k} \right) = 0 \\
   \Rightarrow 8 + ( - 3\lambda ) + 6 = 0 \\
   \Rightarrow 14 + ( - 3\lambda ) = 0 \\
   \Rightarrow ( - 3\lambda ) = 14 \\
   \Rightarrow \lambda = - \dfrac{{14}}{3} \\
\]
∴ The value of \[\lambda \] is \[\dfrac{{ - 14}}{3}\].

Note: The following vector rules of dot product are to be kept in mind always:
\[\hat i.\hat i = 1\]
\[\hat j.\hat j = 1\]
\[\hat k.\hat k = 1\]
\[\hat i.\hat j = 0\]
\[\hat j.\hat k = 0\]
\[\hat k.\hat i = 0\]
Because of these rules we multiplied the same unit vectors only.