Question

How do you find $a=2b+c$ given $b=<6,3>$ and $c=<-4,8>$?

Answer 1

Hint: In the above question, we have been given to vectors $b$ and $c$ in the matrix representation. Firstly we need to convert them into the standard representation in terms of \[\widehat{i}\] and $\widehat{j}$ unit vectors. The third vector $a$ is related to the given vectors $b$ and $c$ by the vector equation $a=2b+c$. On substituting the vectors $b$ and $c$ into the given vector equation, and on adding the corresponding components of the unit vectors \[\widehat{i}\] and $\widehat{j}$, we will get the required vector $a$. Finally we have to write the vector $a$ in the matrix form, as is given in the above question.

Complete step by step solution:
According to the above question, the two vectors given to us are $b=<6,3>$ and $c=<-4,8>$. These vectors have been given in the matrix representation. Since they have two components, they must be expressed in the form of the unit vectors \[\widehat{i}\] and $\widehat{j}$. So they can be expressed in the unit vector representation as follows.
\[\begin{align}
  & \Rightarrow b=6\widehat{i}+3\widehat{j}........\left( i \right) \\
 & \Rightarrow c=-4\widehat{i}+8\widehat{j}......\left( ii \right) \\
\end{align}\]
Now, in the above question, the third vector $a$ is related to the vectors $b$ and $c$ by the relation
$\Rightarrow a=2b+c$
Substituting the equations (i) and (ii) into the above equation, we get
\[\begin{align}
  & \Rightarrow a=2\left( 6\widehat{i}+3\widehat{j} \right)+\left( -4\widehat{i}+8\widehat{j} \right) \\
 & \Rightarrow a=12\widehat{i}+6\widehat{j}-4\widehat{i}+8\widehat{j} \\
 & \Rightarrow a=8\widehat{i}+14\widehat{j} \\
\end{align}\]
The above vector can be expressed in the matrix representation as
$\Rightarrow a=<8,14>$
Hence, the vector $a$ is found as $<8,14>$

Note:
We could directly substitute the matrix forms of the given vectors $b$ and $c$ into the given vector equation $a=2b+c$. But converting them into the standard unit vector form is more convenient. Also, do not forget to convert the unit vector form of the vector $a$ into the matrix form. This is because the unit vectors \[\widehat{i}\] and $\widehat{j}$ were assumed by us and not given in the question.