Question

How do you find \[u+v\] and \[u-v\] given \[u=<2,1>\] and \[v=<1,3>\] ?

Answer 1

Hint: In the given question, we have been asked to add and subtract the vectors and we have given two vectors i.e. \[u=<2,1>\] and\[v=<1,3>\]. In the given vector we have two the components of two directions i.e. ‘x’ and ‘y’ directions. So for adding the vectors first we need to add only the values of x-direction and then we add the values of y-direction and then write it in the given vector form that is in the question. For subtracting the two vectors, we need to subtract the values of x-direction then subtract the values of y-direction and then write it in the vector form. In this way we will get our required solution.

Complete step by step answer:
We have given the two vectors,
\[u=<2,1>\]
\[\Rightarrow v=<1,3>\]
It can be represented in the vector form as;
\[~\overrightarrow{u}=\ <2,1>\]
\[\Rightarrow ~\overrightarrow{v}=\ <1,3>\]
Therefore,the addition of two factors;
\[\overrightarrow{u}+\overrightarrow{v}=\ <2,1>+<1,3>\]
Simplifying the above, we get
\[\overrightarrow{u}+\overrightarrow{v}=\ <2+1,1+3>\]
Adding the components of the given vector, we get
\[\overrightarrow{u}+\overrightarrow{v}=\ <3,4>\]
Thus the sum of two given vectors i.e. \[u=<2,1>\] and \[v=<1,3>\] is \[\overrightarrow{u}+\overrightarrow{v}=\ <3,4>\].
Now,the subtraction of two factors;
\[\overrightarrow{u}-\overrightarrow{v}=\ <2,1>-<1,3>\]
Simplifying the above, we get
\[\overrightarrow{u}-\overrightarrow{v}=\ <2-1,1-3>\]
Adding the components of the given vector, we get
\[\overrightarrow{u}-\overrightarrow{v}=\ <1,-2>\]
Thus the sum of two given vectors i.e. \[u=<2,1>\] and \[v=<1,3>\] is \[\overrightarrow{u}-\overrightarrow{v}=\ <1,-2>\].

Hence,\[u+v\] and \[u-v\] are equals to $\ <3,4>$ and $\ <1,-2>$.

Note:Students need to remember that addition and subtraction of vectors are different from the normally performing addition and subtraction on numbers, it is because a vector contains or represents the x, y and z direction respectively. So while adding or subtracting the given vectors we can only add or subtract the value of the same direction only.