Question

Open the Vectors applet. Suppose $\mathbf{v}=2\mathbf{i}+3\mathbf{j}$and $\mathbf{w}=4\mathbf{i}-3\mathbf{j}$.

1. Draw v and then use the red vector to show $3\mathbf{v}$.
2. Use the red vector to show -v.
3. Draw v and w. Then use the red vector to show $\mathbf{v}+\mathbf{w}$.
4. Use the red vector to show $\mathbf{v}-\mathbf{w}$.
5. Use the red vector to show $2\mathbf{v}+\mathbf{w}$.

Short answer

The graph is as follows,

1. 
2. 
   
   
3. 
   
   
4. 
   
   
5. 

Step-by-step solution

Step 1. Given Information 

We are given two vectors,

$$\begin{gathered} \mathbf{v}=2\mathbf{i}+3\mathbf{j} \\ \mathbf{w}=4\mathbf{i}-3\mathbf{j} \end{gathered}$$

Part(a) Step 1. Drawing the vector v and 3v

The graph is as follows,

Part(b) Step 1. Showing -v

The graph is as follows,

Part(c) Step 1. Drawing v and w and v+w

Adding the vectors, we get

$$\begin{gathered} v+w=2i+3j+4i-3j \\ v+w=6i \end{gathered}$$

The graph is as follows,

Part(d) Step 1. Showing v-w

Subtracting the vectors, we get

$$\begin{gathered} v-w=2i+3j-4i+3j \\ v-w=-2i+6j \end{gathered}$$

The graph is as follows,

Part(e) Step 1. Showing 2v+w

The new vector is given by,

$$\begin{gathered} 2v+w=2(2i+3j)+4i-3j \\ =4i+6j+4i-3j \\ =8i+3j \end{gathered}$$

The graph is as follows,