Question

Find the sum of the vectors \[\overrightarrow{a}{}=\hat{i}-2\hat{j}+\hat{k}\], \[\overrightarrow{b}{}=-2\hat{i}+4\hat{j}+5\hat{k}\] and \[\overrightarrow{c}{}=\hat{i}-6\hat{j}-7\hat{k}\].

Answer 1

Hint: To add the vectors together, we simply have to add their corresponding components that have the same direction only. The resultant will also be a vector.

Complete step-by-step solution
Scalar: A scalar is a quantity that can be fully described by only magnitude alone.
Vector: A vector is a quantity that can be fully described by using both magnitude and a direction.
The following points must be considered while performing vector addition:
The given vectors shall be added geometrically and not algebraically.
Adding vectors means finding the resultant of all the vectors in each direction.
The vectors for which the resultant is to be calculated behave independent to each other, that is every vector behaves as if the other vectors are absent.
Also, the addition of vectors follows the commutative law, so the addition of vectors will be independent of the order of the vectors considered.
   example: \[\overrightarrow{A}{}+\overrightarrow{B}{}=\overrightarrow{B}{}+\overrightarrow{A}{}\].
The vector addition also obeys associative law:
   Mathematically, \[\overrightarrow{A}{}+\left( \overrightarrow{B}{}+\overrightarrow{C}{} \right)=\left( \overrightarrow{A}{}+\overrightarrow{B}{} \right)+\overrightarrow{C}{}\].
The vector addition is also distributive in nature:
Mathematically, \[m\overrightarrow{a}+m\overrightarrow{b}=m\left( \overrightarrow{a}+\overrightarrow{b} \right)\].
The given vectors for addition are:
\[\overrightarrow{a}=\hat{i}-2\hat{j}+\hat{k}\]
\[\overrightarrow{b}{}=-2\hat{i}+4\hat{j}+5\hat{k}\]
\[\overrightarrow{c}{}=\hat{i}-6\hat{j}-7\hat{k}\]
Now to find the sum of these vectors, let us add the corresponding coefficients of \[\hat{i},\hat{j},\hat{k}\] as mentioned below accordingly:
\[\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\left( 1-2+1 \right)\hat{i}+\left( -2+4-6 \right)\hat{j}+\left( 1+5-7 \right)\hat{k}\]
\[\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\left( 0 \right)\hat{i}-\left( 4 \right)\hat{j}-\hat{k}\]
\[\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=-4\hat{j}-\hat{k}\].
So, the sum of the vectors, \[\overrightarrow{a}{}=\hat{i}-2\hat{j}+\hat{k}\], \[\overrightarrow{b}{}=-2\hat{i}+4\hat{j}+5\hat{k}\] and \[\overrightarrow{c}{}=\hat{i}-6\hat{j}-7\hat{k}\] is \[-4\hat{j}-\hat{k}\].
Hence, the answer is \[-4\hat{j}-\hat{k}\].

Note:- We have to be clear that scalar quantities can be added algebraically in a simple manner whereas the vector quantities should be added geometrically only. You must be attentively add only the corresponding components or you may end up with a wrong answer.