# Virtual Math Learning Center

## Practice Problems for Module 1

### Exercises

Directions: You should try to solve each problem first, and then click "Reveal Answer" to check your answer. You can click "Watch Video" if you need help with a problem.

1. Find the vector $$\overrightarrow{AB}$$.

1. ​$$A(-3,4), B (-1,2)$$
2. $$A(0, 0), B(1, 1)$$
3. $$A(−2, 2), B(−1, 3)$$

a) $$\langle 4, -6\rangle$$
b) $$\langle 1, 1\rangle$$
c) $$\langle 1, 1\rangle$$

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2. If $$\mathbf{a}=\langle−1,2\rangle$$ and $$\mathbf{b}=\langle5,3\rangle$$, find $$|\mathbf{a}|,\mathbf{a}+\mathbf{b},\mathbf{a}−\mathbf{b},$$ and $$−3\mathbf{a}+4\mathbf{b}.$$

$$|\mathbf{a}|=\sqrt{5}$$
$$\mathbf{a}+\mathbf{b}=\langle 4,5\rangle$$
$$\mathbf{a}-\mathbf{b}=\langle -6, -1\rangle$$
$$-3\mathbf{a}+4\mathbf{b}=\langle23, 6\rangle$$

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3. If $$|\mathbf{r}| = 2$$, and $$\mathbf{r}$$ makes an angle of $$210^\circ$$ with the positive $$x$$-axis, find the components of the vector $$\mathbf{r}.$$

$$\left\langle -\sqrt{3},-1\right\rangle$$

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4. If $$\mathbf{a} = \langle 3, −4\rangle$$, find a vector with length 10 in the direction of $$\mathbf{a}.$$

$$\left\langle 6,-8 \right\rangle$$

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5. Find $$\mathbf{a}\cdot \mathbf{b}$$.

1. ​$$|\mathbf{a}| = 4$$, $$|\mathbf{b}|=5$$, the angle between $$\mathbf{a}$$ and $$\mathbf{b}$$ is $$\dfrac{\pi}{3}$$.
2. $$\mathbf{a}=\langle -2,-8\rangle$$, $$\mathbf{b}=\langle 6,-4\rangle$$
3. $$\mathbf{a}=\mathbf{i}+\mathbf{j}, \mathbf{b}=\mathbf{i}-2\mathbf{j}$$

a) $$\mathbf{a}\cdot \mathbf{b}=10$$
b) $$\mathbf{a}\cdot \mathbf{b}=20$$
c) $$\mathbf{a}\cdot \mathbf{b}=-1$$

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6. Find the angle between the vectors.

1. $$\mathbf{a}=\langle 6,0\rangle, \mathbf{b}=\langle 5,3\rangle$$
2. $$\mathbf{a}=\langle 3,1\rangle, \mathbf{b}=\langle 2,4\rangle$$

a) $$\theta=\arccos \left(\dfrac{5}{\sqrt{34}}\right)$$

b) $$\theta=\arccos \left(\dfrac{1}{\sqrt{2}}\right)=\dfrac{\pi}{4}=45^\circ$$

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7. Find the values of $$x$$ such that the given vectors are orthogonal.

1. ​$$\langle 4,x\rangle, \langle x, 1\rangle$$
2. $$\langle x, x\rangle, \langle 1, x\rangle$$

a) $$x=0$$
b) $$x=0$$ or $$x=1$$

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8. A force $$\mathbf{F} = \langle−3, 4\rangle$$ is used to move an object from the point $$(0, 2)$$ to the point $$(−3, 3)$$. How much work is done by the force if distance is measured in meters and force is measured in Newtons?​

13 N$$\cdot$$m or J

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9. A boat heads in the direction N30$$^\circ$$E with a speed of 40 mph. The water current is flowing S45$$^\circ$$E with a speed of 6 mph. Find the true speed and direction of the boat.

True Course: $$\langle 3\sqrt{2}, -3\sqrt{2}\rangle$$
Speed: $$\sqrt{ (20+3\sqrt{2})^2 + (20\sqrt{3}-3\sqrt{2})^2}$$
Direction: N$$\theta^\circ$$E, where $$\theta=\arctan\left(\dfrac{20+3\sqrt{2}}{20\sqrt{3}-3\sqrt{2}}\right)$$

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10. Find the vector and scalar projection of $$\langle 4, 8\rangle$$ onto $$\langle 2, 1\rangle.$$

Scalar Projection: $$\dfrac{16}{\sqrt{5}}$$
Vector Projection: $$\left\langle \dfrac{32}{5}, \dfrac{16}{5}\right\rangle$$

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11. Find the vector and scalar projection of $$\langle 2, 1\rangle$$ onto $$\langle 4, 8\rangle.$$

Scalar Projection: $$\dfrac{4}{\sqrt{5}}$$
Vector Projection: $$\left\langle \dfrac{4}{5}, \dfrac{8}{5}\right\rangle$$

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12. Find the distance from the point $$P(2, 1)$$ to the line $$y = 3x + 1.$$

$$\dfrac{6}{\sqrt{10}}$$

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