Question

If \[\overrightarrow P = m\widehat i - 2\widehat j + 2\widehat k\] and \[\overrightarrow Q = 2\widehat i - n\widehat j + \widehat k\] are parallel to each other, then find \[m\] and \[n\].

Answer 1

Hint: The two given vectors \[\overrightarrow P \] and \[\overrightarrow Q \] are given by \[\widehat i\], \[\widehat j\] and \[\widehat k\] components. These vectors \[\overrightarrow P \] and \[\overrightarrow Q \] are parallel to each other. The vectors parallel to each other are known as collinear vectors and they can be written by \[\overrightarrow P = \overrightarrow Q \] . Equating these two vectors will give the values of \[m\] and \[n\].

Formula Used:
When two vectors \[\overrightarrow P \] and \[\overrightarrow Q \] are parallel to each other then they can be written as \[\overrightarrow P = \overrightarrow Q \]

Complete step by step answer:
 The physical quantities which either possess no direction or their direction is not taken into count while adding or multiplying them are called scalars. Magnitude is an essential characteristic of all physical quantities. The physical quantities having both magnitude as well as direction are known as vectors. A vector is denoted by putting an arrowhead over the symbol of the vector. Thus \[\overrightarrow P \] and \[\overrightarrow Q \] represent vectors.
The two given vectors \[\overrightarrow P \] and \[\overrightarrow Q \] are parallel to each other. Therefore, they can be represented as collinear vectors. Thus, we can write
\[\overrightarrow P = \overrightarrow Q \to m\widehat i - 2\widehat j + 2\widehat k = 2\widehat i - n\widehat j + \widehat k\]
Or \[\dfrac{m}{2} = \dfrac{2}{n} = \dfrac{2}{1}\].
This can be individually written as,
\[\dfrac{m}{2} = \dfrac{2}{1}{\text{ or }}\dfrac{2}{n} = \dfrac{2}{1}\].
Equating the denominators to get the value of \[m\].
\[\dfrac{m}{2} = \dfrac{{2 \times 2}}{{1 \times 2}} = \dfrac{4}{2} \\
\Rightarrow m = 4\]
The value of \[n\] is
\[\dfrac{2}{n} = \dfrac{2}{1} \\
\Rightarrow n = 1\]

Therefore, \[m\] and \[n\] are 4 and 1 respectively.

Note:The vectors parallel to each other are known as collinear vectors and they can be written by \[\overrightarrow P = \overrightarrow Q \] . But the vectors that are antiparallel are also referred to as collinear vectors and can be written by \[\overrightarrow P = - \overrightarrow Q \]. Note that any two vectors are only said to be equal if their directions are the same and the magnitudes are equal.