Question

Find the sum of the following vectors :
$\overrightarrow a = \widehat i - 2\widehat j,\overrightarrow b = 2\widehat i - 3\widehat j,\overrightarrow c = 2\widehat i + 3\widehat k$

Answer 1

Hint: Here $\widehat i$ represent the given vector in x –direction, $\widehat j$ represent the given vector in y – direction and $\widehat k$ represent the given vector in z-direction respectively. So from the property of the vector addition we know that the vectors of the same coordinates can be added together to get the required resultant addition or sum of the vectors.

Complete step by step answer:
We know that $\widehat i$ represent the given vector in x –direction, $\widehat j$ represent the given vector in y – direction and $\widehat k$ represent the given vector in z-direction respectively. So from the property of the vector addition we know that the vectors of the same coordinates can be added together to get the required resultant addition or sum of the vectors.
Let us consider that the resultant of the vector addition to be represented as $\overrightarrow s $
Here $\overrightarrow s $ can be expressed as the sum of $\overrightarrow a $, $\overrightarrow b $ and $\overrightarrow c $ respectively.
Hence, we can write mathematically that $\overrightarrow s = \overrightarrow a + \overrightarrow b + \overrightarrow c $
We are given that $\overrightarrow a = \widehat i - 2\widehat j,\overrightarrow b = 2\widehat i - 3\widehat j,\overrightarrow c = 2\widehat i + 3\widehat k$
So further substituting the given values, we can express the equation as
$\overrightarrow s $$ = (\widehat i - 2\widehat j) + (2\widehat i - 3\widehat j) + (2\widehat i + 3\widehat k)$
Now when we add the $\widehat i$,$\widehat j$and $\widehat k$ vectors together , the equation becomes
$\overrightarrow s $$ = (\widehat i + 2\widehat i + 2\widehat i) + ( - 2\widehat j - 3\widehat j) + 3\widehat k$
Now further simplifying the equation becomes
$\overrightarrow s = (5\widehat i) + ( - 5\widehat j) + 3\widehat k$
Which we can further express as
$\overrightarrow s = 5\widehat i - 5\widehat j + 3\widehat k$
Hence we get the sum of the given vectors as $5\widehat i - 5\widehat j + 3\widehat k$.

Note:
Here the student may commit a mistake of adding the $\widehat i$ vector terms with the $\widehat j$ vector terms and similarly adding the unlike coordinate vector terms to get the required sum. Hence the student must beforehand group the similar coordinate vectors and sum it up to get the required summation of the resultant vectors.