Question

Given vectors $u=<4,1>$ and $v=<1,3>$ , how would I determine the quantity $3u^*-2v$ ?

Answer 1

Hint:Here in this question we have been given two vectors $u=<4,1>$ and $v=<1,3>$ and asked to determine the quantity $3u^*-2v$ . We know that $u^*$ is the complement of $u$ and it is given as $u^*=<4,-1>$ since the value of the vector $u$ is given as $<4,1>$ .

Complete step by step answer:
Now considering from the question we have been given two vectors $u=<4,1>$ and $v=<1,3>$ and asked to determine the quantity $3u^*-2v$ .
From the basic concepts of vectors we know that $u^*$ is the complement of $u$ and it is given as $u^*=<4,-1>$ since the value of the vector $u$ is given as $<4,1>$ .
Now by substituting the given values of vectors in the given expression $3u^*-2v$ we will have $\Rightarrow 3\left( 4,-1 \right)-2\left( 1,3 \right)$ .
Now by performing arithmetic operations of vectors on this expression we will have $\Rightarrow \left( 12,-3 \right)-\left( 2,6 \right)$ .
Now by further simplifying this expression we will have $\Rightarrow \left( 12-2,-3-6 \right)$ .
By further simplifying this expression we will have $\Rightarrow \left( 10,-9 \right)$ .
Therefore we can conclude that when we have been given two vectors $u=<4,1>$ and $v=<1,3>$ then the value of the quantity $3u^*-2v$ will be given as $\left( 10,-9 \right)$ .

Note: While answering questions of this type we should be sure with the concept that we are applying and calculations that we are performing in between. This is a very simple and easy question which can be answered accurately in a short span of time. Someone can make a calculation mistake unintentionally and consider it as
$\begin{align}
  & 3u^*-2v=3\left( 4,-1 \right)-2\left( 1,3 \right) \\
 & \Rightarrow \left( 12,-3 \right)-\left( 2,6 \right)=\left( 12-2,-3-6 \right) \\
 & \Rightarrow \left( 9,-9 \right) \\
\end{align}$
which leads us to end up having a wrong conclusion.