Question

For what the value of $\lambda$ are the vectors $\overrightarrow a {\text{ and }}\overrightarrow b$ perpendicular to each other? Where:
$\overrightarrow a = \lambda \widehat i + 2\widehat j + \widehat k{\text{ and }}\overrightarrow b = 4\widehat i - 9\widehat j + 2\widehat k$

Answer 1

Hint-In this question, two perpendicular vectors are given to us, use the fact that two vectors are perpendicular if and only if their scalar product is equal to zero. Simply equate both the sides and get the answer.

Complete step-by-step answer:
Two given vectors are
$\overrightarrow a = \lambda \widehat i + 2\widehat j + \widehat k \\\ \overrightarrow b = 4\widehat i - 9\widehat j + 2\widehat k \\\$
$\overrightarrow a {\text{ and }}\overrightarrow b$are perpendicular to each other
$\Rightarrow \overrightarrow a .\overrightarrow b = 0$
$(\lambda \widehat i + 2\widehat j + \widehat k{\text{)}}{\text{.( }}4\widehat i - 9\widehat j + 2\widehat k) = 0 \\\ 4\lambda - 18 + 2 = 0 \\\ 4\lambda - 16 = 0 \\\ \lambda = \dfrac{{16}}{4} \\\ \lambda = 4 \\\$
Hence when $\lambda$ is 4, $\overrightarrow a {\text{ is perpendicular to }}\overrightarrow b$.
Note-For these types of questions, the key concept is that i, j and k are the unit vectors along x-axis, y-axis and z-axis respectively and hence are also perpendicular to each other. Dot product between these unit vectors is equal to zero e.g. i.j = 0, j.k=0, k.i=0, etc.