Question

Let $\mathit{A}\mathbf{=}\mathbf{2}\mathit{i}\mathbf{+}\mathbf{3}\mathit{j}$and $\mathit{B}\mathbf{=}\mathbf{4}\mathit{i}\mathbf{-}\mathbf{4}\mathit{j}$. Show graphically, and find algebraically, the vectors $\mathbf{-}\mathit{A}\mathbf{,}\mathbf{3}\mathit{B}\mathbf{,}\mathit{A}\mathbf{-}\mathit{B}\mathbf{,}\mathit{B}\mathbf{+}\mathbf{2}\mathit{A}\mathbf{,}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{(}\mathit{A}\mathbf{+}\mathit{B}\mathbf{)}$.

Short answer

We use the rules of adding vectors graphically to solve this problem.

Step-by-step solution

Concept of vector

There are two ways to get the sum of two vectors. One is by the parallelogram law and the other is to find $\mathit{A}\mathbf{+}\mathit{B}$, place the tail of$\mathit{B}$at the head of$\mathit{A}$and draw the vector from the tail of$\mathit{A}$to the head of$\mathit{B}$.

 Step 2: Calculating the vectors

Given: $A=2i+3j$and $B=4i-4j$.

$$\begin{gathered} -A=-(2i+3j) \\ =-2i-3j \end{gathered}$$

which is shown in the following graph

$$\begin{gathered} 3B=3\left(4i-4j\right) \\ =12i-12j \end{gathered}$$

As shown in the following graph

$\bullet (A-B)$vector:

$$\begin{gathered} A-B=(2i+3j)-(4i-4j) \\ =-2i+7j \end{gathered}$$

As shown in Graph

$\bullet (B+2A)$vector:

$$\begin{gathered} B+2A=4i-4j+2\left(2i+3j\right) \\ =4i-4j+4i+6j \\ =8i+2j \end{gathered}$$

Graph of the vector is as follows

$\bullet \left(\frac{1}{2}(A+B)\right)$vector:

$$\begin{gathered} \frac{1}{2}(A+B)=\frac{1}{2}(2i+3j+4i-4j) \\ =3i-\frac{1}{2}j \end{gathered}$$

The graph of the vector is shown below.

We use the rules of adding vectors graphically to solve this problem.