Precalculus Practice Problems: Vectors, Matrices, and Statistics

Vectors 📏

Problem 1: Given vectors $\vec{a} = \langle 2, -3 \rangle$ and $\vec{b} = \langle -1, 5 \rangle$, find $\vec{a} + \vec{b}$ and $3\vec{a} - 2\vec{b}$.

Solution:

• $\vec{a} + \vec{b} = \langle 2 + (-1), -3 + 5 \rangle = \langle 1, 2 \rangle$
• $3\vec{a} - 2\vec{b} = 3\langle 2, -3 \rangle - 2\langle -1, 5 \rangle = \langle 6, -9 \rangle - \langle -2, 10 \rangle = \langle 8, -19 \rangle$

Problem 2: Find the unit vector in the direction of $\vec{v} = \langle 4, -3 \rangle$.

Solution:

1. Find the magnitude of $\vec{v}$: $|\vec{v}| = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$
2. Divide $\vec{v}$ by its magnitude: $\hat{u} = \frac{\vec{v}}{|\vec{v}|} = \frac{\langle 4, -3 \rangle}{5} = \langle \frac{4}{5}, -\frac{3}{5} \rangle$

Matrices 🔢

Problem 1: Given matrices $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$, find $A + B$ and $AB$.

Solution:

• $A + B = \begin{bmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}$
• $AB = \begin{bmatrix} (1)(5)+(2)(7) & (1)(6)+(2)(8) \\ (3)(5)+(4)(7) & (3)(6)+(4)(8) \end{bmatrix} = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$

Problem 2: Find the determinant of the matrix $C = \begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix}$.

Solution:

$\det(C) = (2)(3) - (1)(4) = 6 - 4 = 2$

Statistics 📊

Problem 1: Given the data set: 5, 8, 10, 12, 15. Find the mean, median, and standard deviation.

Solution:

• Mean: $\frac{5+8+10+12+15}{5} = \frac{50}{5} = 10$
• Median: 10 (the middle value when the data is sorted)
• Standard Deviation:

import numpy as np

data = [5, 8, 10, 12, 15]
std_dev = np.std(data)
print(std_dev)

Output: 3.41

Problem 2: A bag contains 3 red balls and 5 blue balls. What is the probability of drawing a red ball?

Solution:

Probability = (Number of red balls) / (Total number of balls) = $3 / (3+5) = 3/8$