Question

Find a vector of magnitude $18$ which is perpendicular to both the vectors, $4\widehat i - \widehat j + 3\widehat k$ and $ - 2\widehat i + \widehat j - 2\widehat k$.
A) $ - 8\widehat i + 12\widehat j + 12\widehat k$
B) $ - 7\widehat i + 12\widehat j + 12\widehat k$
C) $ - 6\widehat i + 12\widehat j + 12\widehat k$
D) $ - 9\widehat i + 12\widehat j + 12\widehat k$

Answer 1

Hint: Recall that vector is a quantity that has both magnitude and direction. It is also known as direction vector. A vector can be converted into a unit vector if it is divided by the magnitude of the vector given. The symbol ‘cap’ or ‘hat’ is used to represent a unit vector. But they are dimensionless quantities and they also do not have any units.

Complete step by step solution:
A vector is said to be perpendicular to the given two vectors if it is the result of the cross product of two vectors given. Any vector in the space can be expressed as a linear combination of the unit vectors.
Given that the two vectors are
$\overrightarrow a = 4\widehat i - \widehat j + 3\widehat k$
And $\overrightarrow b = - 2\widehat i + \widehat j - 2\widehat k$
Their cross product will be represented as
$\overrightarrow a \times \overrightarrow b = \widehat i(2 - 3) - \widehat j( - 8 + 6) + \widehat k(4 - 2)$
$ \Rightarrow \overrightarrow a \times \overrightarrow b = - \widehat i + 2\widehat j + 2\widehat k$
The magnitude of the cross product of the two vectors will be written as
$\left| {\overrightarrow a \times \overrightarrow b } \right| = \sqrt {{{( - 1)}^2} + {{(2)}^2} + {{(2)}^2}} $
$\left| {\overrightarrow a \times \overrightarrow b } \right| = \sqrt {1 + 4 + 4} $
$ \Rightarrow \left| {\overrightarrow a \times \overrightarrow b } \right| = \sqrt 9 = 3$
Therefore, the required vector that is perpendicular to the vectors is given by
$ \Rightarrow \dfrac{{18}}{3}( - \widehat i + 2\widehat j + 2\widehat k)$
$ \Rightarrow 6( - \widehat i + 2\widehat j + 2\widehat k)$
$ \Rightarrow - 6\widehat i + 12\widehat j + 12\widehat k$
The vector that is perpendicular to both the vectors is $ - 6\widehat i + 12\widehat j + 12\widehat k.$

Option C is the right answer.

Note: It is important to remember that a unit vector has always a magnitude of 1. Also a unit vector has the same direction as that of the given vector. When a vector is parallel to the given vectors, then it can be known by the dot product of two vectors given. A unit normal vector is a vector that is perpendicular to the surface at some given point. Such a vector is also known as normal.