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

How do you find u+vu+v and u−vu-v given u=<2,1>u=<2,1> and v=<1,3>v=<1,3> ?

Explanation

Solution

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>u=<2,1> andv=<1,3>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>u=<2,1>
⇒v=<1,3>\Rightarrow v=<1,3>
It can be represented in the vector form as;
 u→= <2,1>~\overrightarrow{u}=\ <2,1>
⇒ v→= <1,3>\Rightarrow ~\overrightarrow{v}=\ <1,3>
Therefore,the addition of two factors;
u→+v→= <2,1>+<1,3>\overrightarrow{u}+\overrightarrow{v}=\ <2,1>+<1,3>
Simplifying the above, we get
u→+v→= <2+1,1+3>\overrightarrow{u}+\overrightarrow{v}=\ <2+1,1+3>
Adding the components of the given vector, we get
u→+v→= <3,4>\overrightarrow{u}+\overrightarrow{v}=\ <3,4>
Thus the sum of two given vectors i.e. u=<2,1>u=<2,1> and v=<1,3>v=<1,3> is u→+v→= <3,4>\overrightarrow{u}+\overrightarrow{v}=\ <3,4>.
Now,the subtraction of two factors;
u→−v→= <2,1>−<1,3>\overrightarrow{u}-\overrightarrow{v}=\ <2,1>-<1,3>
Simplifying the above, we get
u→−v→= <2−1,1−3>\overrightarrow{u}-\overrightarrow{v}=\ <2-1,1-3>
Adding the components of the given vector, we get
u→−v→= <1,−2>\overrightarrow{u}-\overrightarrow{v}=\ <1,-2>
Thus the sum of two given vectors i.e. u=<2,1>u=<2,1> and v=<1,3>v=<1,3> is u→−v→= <1,−2>\overrightarrow{u}-\overrightarrow{v}=\ <1,-2>.

Hence,u+vu+v and u−vu-v are equals to  <3,4>\ <3,4> and  <1,−2>\ <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.