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.