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 \(\overrightarrow{AB}\).
- \(A(-3,4), B (-1,2)\)
- \(A(0, 0), B(1, 1)\)
- \(A(−2, 2), B(−1, 3)\)
a) \( \langle 4, -6\rangle \)
b) \(\langle 1, 1\rangle\)
c) \(\langle 1, 1\rangle\)
b) \(\langle 1, 1\rangle\)
c) \(\langle 1, 1\rangle\)
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 \(−3\mathbf{a}+4\mathbf{b}.\)
\(|\mathbf{a}|=\sqrt{5}\)
\(\mathbf{a}+\mathbf{b}=\langle 4,5\rangle\)
\(\mathbf{a}-\mathbf{b}=\langle -6, -1\rangle\)
\(-3\mathbf{a}+4\mathbf{b}=\langle23, 6\rangle\)
\(\mathbf{a}+\mathbf{b}=\langle 4,5\rangle\)
\(\mathbf{a}-\mathbf{b}=\langle -6, -1\rangle\)
\(-3\mathbf{a}+4\mathbf{b}=\langle23, 6\rangle\)
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 \(\mathbf{r}.\)
\( \left\langle -\sqrt{3},-1\right\rangle\)
4. If \( \mathbf{a} = \langle 3, −4\rangle\), find a vector with length 10 in the direction of \(\mathbf{a}.\)
\( \left\langle 6,-8 \right\rangle\)
5. Find \(\mathbf{a}\cdot \mathbf{b}\).
- \(|\mathbf{a}| = 4\), \(|\mathbf{b}|=5\), the angle between \(\mathbf{a}\) and \(\mathbf{b}\) is \(\dfrac{\pi}{3}\).
- \(\mathbf{a}=\langle -2,-8\rangle\), \(\mathbf{b}=\langle 6,-4\rangle\)
- \(\mathbf{a}=\mathbf{i}+\mathbf{j}, \mathbf{b}=\mathbf{i}-2\mathbf{j}\)
a) \( \mathbf{a}\cdot \mathbf{b}=10\)
b) \( \mathbf{a}\cdot \mathbf{b}=20\)
c) \( \mathbf{a}\cdot \mathbf{b}=-1\)
b) \( \mathbf{a}\cdot \mathbf{b}=20\)
c) \( \mathbf{a}\cdot \mathbf{b}=-1\)
6. Find the angle between the vectors.
- \(\mathbf{a}=\langle 6,0\rangle, \mathbf{b}=\langle 5,3\rangle\)
- \(\mathbf{a}=\langle 3,1\rangle, \mathbf{b}=\langle 2,4\rangle\)
a) \(\theta=\arccos \left(\dfrac{5}{\sqrt{34}}\right)\)
b) \(\theta=\arccos \left(\dfrac{1}{\sqrt{2}}\right)=\dfrac{\pi}{4}=45^\circ\)
b) \(\theta=\arccos \left(\dfrac{1}{\sqrt{2}}\right)=\dfrac{\pi}{4}=45^\circ\)
7. Find the values of \(x\) such that the given vectors are orthogonal.
- \(\langle 4,x\rangle, \langle x, 1\rangle\)
- \(\langle x, x\rangle, \langle 1, x\rangle\)
a) \(x=0\)
b) \(x=0\) or \(x=1\)
b) \(x=0\) or \(x=1\)
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
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 \(\theta=\arctan\left(\dfrac{20+3\sqrt{2}}{20\sqrt{3}-3\sqrt{2}}\right)\)
Speed: \(\sqrt{ (20+3\sqrt{2})^2 + (20\sqrt{3}-3\sqrt{2})^2}\)
Direction: N\(\theta^\circ\)E, where \(\theta=\arctan\left(\dfrac{20+3\sqrt{2}}{20\sqrt{3}-3\sqrt{2}}\right)\)
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: \(\left\langle \dfrac{32}{5}, \dfrac{16}{5}\right\rangle\)
Vector Projection: \(\left\langle \dfrac{32}{5}, \dfrac{16}{5}\right\rangle\)
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: \(\left\langle \dfrac{4}{5}, \dfrac{8}{5}\right\rangle\)
Vector Projection: \(\left\langle \dfrac{4}{5}, \dfrac{8}{5}\right\rangle\)
12. Find the distance from the point \(P(2, 1)\) to the line \(y = 3x + 1.\)
\(\dfrac{6}{\sqrt{10}}\)