Question

Operations with Vectors

Find $2\mathbf{u},-3\mathbf{v},\mathbf{u}+\mathbf{v}\&3u-4v$for the given vectors $u$and $v$

$\mathbf{u}\mathbf{=}\mathbf{i}\mathbf{+}\mathbf{j}\mathbf{,}\mathbf{v}\mathbf{=}\mathbf{i}\mathbf{-}\mathbf{j}$

Short answer

The required values are-

$\begin{array}{l}2u=2i+2j\\ -3v=-3i+3j\\ u+v=2i\\ 3u-4v=-i+7j\end{array}$

Step-by-step solution

Step 1. Given information

Two points are given as $u=i+j\&v=i-j$

Step 2. Concept used

Vectors can be manipulated in the same way as algebraic expression as-

$\begin{array}{l}(ai+bj)+(ci+dj)=(a+c)i+(b+d)j\\ (ai+bj)-(ci+dj)=(a-c)i+(b-d)j\\ x(ai+bj)=\left(xa\right)i+\left(xb\right)j\end{array}$$\begin{array}{l}(ai+bj)+(ci+dj)=(a+c)i+(b+d)j\\ (ai+bj)-(ci+dj)=(a-c)i+(b-d)j\\ x(ai+bj)=\left(xa\right)i+\left(xb\right)j\end{array}$

Use vector algebraic operations accordingly to get the value for each condition.

Step 3. Calculation

Substitute the values of and in the required terms,

$\begin{array}{l}2u=2(i+j)\\ 2u=2i+2j,\\ -3v=-3(i-j)\\ -3v=-3i+3j,\\ u+v=i+j+i-j\\ u+v=2i,\\ 3u-4v=3(i+j)-4(i-j)\\ =3i+3j-4i+4j,\\ 3u-4v=-i+7j\end{array}$