Question

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

Answer 1

Hint: We have given the three vectors a, b and c and we are asked to find the sum of these three vectors. We are going to find the sum of these three vectors by adding the coefficients of $\overset{\wedge }{\mathop{i}}\,$ of a, b and c together then $\overset{\wedge }{\mathop{j}}\,$ of a, b and c together and $\overset{\wedge }{\mathop{k}}\,$ of a, b and c together. And that’s how we will get the sum of three vectors a, b and c.

Complete step by step answer:
The three vectors a, b and c given in the above problem is:
\[\overrightarrow{a}=\overset{\wedge }{\mathop{i}}\,-2\overset{\wedge }{\mathop{j}}\,+\overset{\wedge }{\mathop{k}}\,,\overrightarrow{b}=\overset{\wedge }{\mathop{-2i}}\,+4\overset{\wedge }{\mathop{j}}\,+5\overset{\wedge }{\mathop{k}}\,,\overrightarrow{c}=\overset{\wedge }{\mathop{i}}\,-6\overset{\wedge }{\mathop{j}}\,+-7\overset{\wedge }{\mathop{k}}\,\]
We are asked to find the summation of the above three vectors which we are going to add by adding the coefficients of $\overset{\wedge }{\mathop{i}}\,$ of a, b and c together then $\overset{\wedge }{\mathop{j}}\,$ of a, b and c together and $\overset{\wedge }{\mathop{k}}\,$ of a, b and c together.
Adding the coefficients of $\overset{\wedge }{\mathop{i}}\,$ of a, b and c together we get,
\[\begin{align}
  & \overset{\wedge }{\mathop{i}}\,-2\overset{\wedge }{\mathop{i}}\,+\overset{\wedge }{\mathop{i}}\, \\
 & =2\overset{\wedge }{\mathop{i}}\,-2\overset{\wedge }{\mathop{i}}\, \\
 & =0.\overset{\wedge }{\mathop{i}}\, \\
\end{align}\]
Adding the coefficients of $\overset{\wedge }{\mathop{j}}\,$ of a, b and c together we get,
\[\begin{align}
  & -2\overset{\wedge }{\mathop{j}}\,+4\overset{\wedge }{\mathop{j}}\,-6\overset{\wedge }{\mathop{j}}\, \\
 & =4\overset{\wedge }{\mathop{j}}\,-8\overset{\wedge }{\mathop{j}}\, \\
 & =-4\overset{\wedge }{\mathop{j}}\, \\
\end{align}\]
Adding the coefficients of $\overset{\wedge }{\mathop{k}}\,$ of a, b and c together we get,
\[\begin{align}
  & \overset{\wedge }{\mathop{k}}\,+5\overset{\wedge }{\mathop{k}}\,-7\overset{\wedge }{\mathop{k}}\, \\
 & =6\overset{\wedge }{\mathop{k}}\,-7\overset{\wedge }{\mathop{k}}\, \\
 & =-\overset{\wedge }{\mathop{k}}\, \\
\end{align}\]
Now, adding all the three unit vectors that we have solved above we get,
\[\begin{align}
  & 0-4\overset{\wedge }{\mathop{j}}\,-\overset{\wedge }{\mathop{k}}\, \\
 & =-4\overset{\wedge }{\mathop{j}}\,-\overset{\wedge }{\mathop{k}}\, \\
\end{align}\]

Hence, we have got the addition of three vectors a, b and c as:
\[-4\overset{\wedge }{\mathop{j}}\,-\overset{\wedge }{\mathop{k}}\,\]

Note: While adding the three vectors, the plausible mistake that could happen in solving the above problem is wrongly taking the coefficients of the respective unit vectors. For e.g., while adding the coefficients of $\overset{\wedge }{\mathop{i}}\,$ of a, b and c together, you mistakenly took the coefficients of $\overset{\wedge }{\mathop{j}}\,or \overset{\wedge }{\mathop{k}}\,$ because the way a, b and c vectors are written in the above solution picking the wrong coefficient is pretty high.
The remedy to this problem is writing the three vectors as follows:
\[\begin{align}
  & \overrightarrow{a}=\overset{\wedge }{\mathop{i}}\,-2\overset{\wedge }{\mathop{j}}\,+\overset{\wedge }{\mathop{k}}\,, \\
 & \overrightarrow{b}=\overset{\wedge }{\mathop{-2i}}\,+4\overset{\wedge }{\mathop{j}}\,+5\overset{\wedge }{\mathop{k}}\,, \\
 & \overrightarrow{c}=\overset{\wedge }{\mathop{i}}\,-6\overset{\wedge }{\mathop{j}}\,+-7\overset{\wedge }{\mathop{k}}\, \\
\end{align}\]
Now, you can clearly see the coefficients of the unit vectors and can prevent the wrong picking of the coefficients.