Question

Find the sum of the vectors \[\overset{\to }{\mathop{a}}\,=\hat{i}-2\hat{j}+\hat{k},\] \[\overset{\to }{\mathop{b}}\,=-2\hat{i}+4\hat{j}+5\hat{k}\] and \[\overset{\to }{\mathop{c}}\,=\hat{i}-6\hat{j}-7\hat{k}\].
(a) \[-4\hat{j}-\hat{k}\]
(b) \[\hat{i}-4\hat{j}-\hat{k}\]
(c) \[\hat{i}+5\hat{j}+2\hat{k}\]
(d) \[-3\hat{i}-2\hat{j}+\hat{k}\]

Answer 1

Hint: The result obtained after adding the vectors is called resultant. In order to solve these types of questions, we are given three vectors and we are asked to find the sum of these three vectors. The addition of vectors is a simple addition. Here, in order to find the sum of the vectors, firstly we will add all the vectors at a place, then we will simply add the coefficients of unit vectors, i.e., $\hat{i},\hat{j},\hat{k}$. Generally, the vector addition is done by adding the tail of a vector to the nose of the previous vector, and finally the resultant vector is obtained.

Complete step by step answer:
Here, the given vectors are as follows:
\[\overset{\to }{\mathop{a}}\,=\hat{i}-2\hat{j}+\hat{k}.....................(i)\]
\[\overset{\to }{\mathop{b}}\,=-2\hat{i}+4\hat{j}+5\hat{k}.............(ii)\] and
\[\overset{\to }{\mathop{c}}\,=\hat{i}-6\hat{j}-7\hat{k}...........(iii)\],
We have to find the sum of the vectors in (i), (ii) and (iii). The addition of vectors is a simple addition. Here, in order to find the sum of the vectors, firstly we will add all the vectors at a place, then we will simply add the coefficients of unit vectors, i.e., $\hat{i},\hat{j},\hat{k}$.
So, on adding these vectors, we get,
\[\Rightarrow \vec{a}+\vec{b}+\vec{c}=\left( \hat{i}-2\hat{j}+\hat{k} \right)+\left( -2\hat{i}+4\hat{j}+5\hat{k} \right)+\left( \hat{i}-6\hat{j}-7\hat{k} \right)\]
Now, we open the braces and so that we can group together the unit vectors of one type, so, we get,
\[\Rightarrow \vec{a}+\vec{b}+\vec{c}=\hat{i}-2\hat{j}+\hat{k}-2\hat{i}+4\hat{j}+5\hat{k}+\hat{i}-6\hat{j}-7\hat{k}\]
Then, Grouping same type of unit vectors at a place, we get,
\[\Rightarrow \vec{a}+\vec{b}+\vec{c}=\left( \hat{i}-2\hat{i}+\hat{i} \right)+\left( -2\hat{j}+4\hat{j}-6\hat{j} \right)+\left( \hat{k}+5\hat{k}-7\hat{k} \right)\]
Finally, adding the coefficients of the respective unit vectors, we get the required sum of vectors,
\[\Rightarrow \vec{a}+\vec{b}+\vec{c}=0\hat{i}+\left( -4\hat{j} \right)+\left( -\hat{k} \right)\]
\[\therefore \vec{a}+\vec{b}+\vec{c}=-4\hat{j}-\hat{k}\]
Hence, the sum of the given vectors is \[-4\hat{j}-\hat{k}\].

So, the correct answer is “Option A”.

Note: Students often cannot differentiate between the unit vectors and add up different types of unit vectors. So, we should be careful while adding up the coefficients of the unit vectors. Besides, Students need to be careful while opening the braces and grouping similar types of vectors. This will ensure that students add the respective coefficients. We need to be careful about the signs of the respective coefficients during the calculation.