7.1: Vectors in the Plane

1. Sketch vector $\mathbf{v} = (-1, 3)$ in standard position, then sketch vector $\mathbf{v}$ again with a different initial point.
    
2. Sketch vector $\mathbf{a} = 2\mathbf{i} + 4\mathbf{j}$ in standard position, then sketch vector $\mathbf{a}$ again with a different initial point.
    
3. Sketch vector $\mathbf{u} = \left(\dfrac{9}{2}, -4\right)$ in standard position, then sketch vector $\mathbf{u}$ again with a different initial point.
    
4. Sketch vector $\mathbf{w} = -\dfrac{3}{2}\mathbf{j}$ in standard position, then sketch vector $\mathbf{w}$ again with a different initial point.
    
5. Consider the points $A(2, 0)$ and $B(-2, 1)$. Express the vector $\vec{AB}$ in component form. Plot the points and the vector.
    
6. Consider the points $P(3, 2)$ and $Q(-4, -3)$. Express the vector $\vec{QP}$ in terms of the standard unit vectors. Plot the points and the vector.
    
7. Perform the vector arithmetic. Express your answer in component form.
   $$\begin{equation*} (-4, 2) + (-9, -9)   \end{equation*}$$
    
8. Perform the vector arithmetic. Express you answer in terms of the standard unit vectors.
   $$\begin{equation*} (\mathbf{i} - 2\mathbf{j}) + (6\mathbf{i} + \mathbf{j}) \end{equation*}$$
    
9. Vectors $\mathbf{u}$ and $\mathbf{v}$ are plotted tip-to-tail below. Using the plot, sketch $\mathbf{w} = \mathbf{u}+\mathbf{v}$.
   
   
    
10. Perform the vector arithmetic. Express your answer in component form.
    $$\begin{equation*}  (-3, 3) - (-8, 2) \end{equation*}$$
     
11. Perform the vector arithmetic. Express you answer in terms of the standard unit vectors.
    $$\begin{equation*}  (-6\mathbf{i} + 8\mathbf{j}) - (-9\mathbf{i})  \end{equation*}$$
     
12. Vectors $\mathbf{u}$ and $\mathbf{v}$ are plotted below with the same initial point. Using the plot, sketch vector $\mathbf{w} = \mathbf{u}-\mathbf{v}$.
    
    
     
13. Let vector $\mathbf{u} = (3, -2)$. Calculate $-3\mathbf{u}$ and express your answer in component form. Plot $\mathbf{u}$ and $-3\mathbf{u}$ on the same coordinate plane.
     
14. Let vector $\mathbf{b} = 5\mathbf{i} - 4\mathbf{j}$. Calculate $\dfrac{1}{2}\mathbf{b}$ and express your answer in terms of the standard unit vectors. Plot $\mathbf{b}$ and $\dfrac{1}{2}\mathbf{b}$ on the same coordinate plane.
     
15. Let vector $\mathbf{u} = (-2, 2)$, vector $\mathbf{v} = (-4, 5)$, and vector $\mathbf{w} = (-2, -5)$. Calculate $\mathbf{r} = -2\mathbf{w} - 5(\mathbf{v} - 3\mathbf{u})$. Express your answer in component form.
     
16. Let vector $\mathbf{u} = 4\mathbf{i} + 2\mathbf{j}$, vector $\mathbf{v} = -\mathbf{i} + \mathbf{j}$, and vector $\mathbf{w} = -5\mathbf{i} - \mathbf{j}$. Calculate $\mathbf{r} = 3(\mathbf{v} - 5\mathbf{u}) - 4\mathbf{w}$. Express your answer in terms of the standard unit vectors.
     
17. Find the norm of vector $\mathbf{v} = (-2, 4)$.
     
18. Find the magnitude of vector $\mathbf{w} = 6\mathbf{i} - 3\mathbf{j}$.
     
19. Find a unit vector $\mathbf{u}$ pointing in the same direction as vector $\mathbf{v} = (-10, -2)$. Express $\mathbf{u}$ in component form.
     
20. Normalize vector $\mathbf{v} = -6\mathbf{i} + 8\mathbf{j}$. Express your answer in terms of the standard unit vectors.
     
21. Find a vector $\mathbf{a}$ in the same direction as vector $\mathbf{b} = (8, 9)$ with magnitude $\|\mathbf{a}\| = 2$. Express your answer in component form.
     
22. Find a vector $\mathbf{u}$ in the same direction as vector $\mathbf{v} = \mathbf{i} - 7\mathbf{j}$ with magnitude $\|\mathbf{u}\| = 3$. Express your answer in terms of the standard unit vectors.
     
23. Find vector $\mathbf{w}$ so that $\|\mathbf{w}\| = 3$ and the angle between $\mathbf{w}$ and the positive $x$-axis is $\theta = \dfrac{5\pi}{4}$. Express your answer in component form.
     
24. Find vector $\mathbf{v}$ so that $\|\mathbf{v}\| = 10$ and the angle between $\mathbf{v}$ and the positive $x$-axis is $\theta = \dfrac{2\pi}{3}$. Express your answer in
    terms of the standard unit vectors.
     
25. Find the angle $\theta \in [0, 2\pi)$ that vector $\mathbf{u} = (-4\sqrt{2}, 4\sqrt{2})$ makes with the positive $x$-axis.
     
26. Find the angle $\theta \in [0, 2\pi)$ that vector $\mathbf{a} = -2\sqrt{3}\mathbf{i} - 2\mathbf{j}$ makes with the positive $x$-axis.