Question 1

Which of the following vectors is orthogonal to \(\vec{v} = \lbrace -1,3,0 \rbrace\) and has a magnitude of \(\sqrt{19}\)?

We will test each answer choice below to determine orthogonality and determine which vector has the specified magnitude.

First, we will definte our vector \(\vec{v}\) and magnitude of \(\sqrt{19}\).

our_vector <- c(-1,3,0)
our_magnitude <- sqrt(19)
cat("Our vector:",our_vector,"\n")
## Our vector: -1 3 0
cat("Our magnitude:",our_magnitude,"\n")
## Our magnitude: 4.358899

A. \(\lbrace 3,1,0 \rbrace\)

# install.packages("pracma")
library(pracma)
A <- c(3,1,0)
if (dot(our_vector,A) == 0 & Norm(A) == our_magnitude) {
  cat("Answer choice A is correct.")
} else {
  cat("Answer choice A is incorrect.","\n")
  cat("Dot Product:",dot(our_vector,A),"\n")
  cat("Magnitude:",Norm(A),"\n")
}
## Answer choice A is incorrect. 
## Dot Product: 0 
## Magnitude: 3.162278

B. \(\lbrace 1,3,3 \rbrace\)

# install.packages("pracma")
library(pracma)
B <- c(1,3,3)
if (dot(our_vector,B) == 0 & Norm(B) == our_magnitude) {
  cat("Answer choice B is correct.")
} else {
  cat("Answer choice B is incorrect.","\n")
  cat("Dot Product:",dot(our_vector,B),"\n")
  cat("Magnitude:",Norm(B),"\n")
}
## Answer choice B is incorrect. 
## Dot Product: 8 
## Magnitude: 4.358899

C. \(\lbrace 3,1,3 \rbrace\)

# install.packages("pracma")
library(pracma)
C <- c(3,1,3)
if (dot(our_vector,C) == 0 & Norm(C) == our_magnitude) {
  cat("Answer choice C is correct.")
} else {
  cat("Answer choice C is incorrect.","\n")
  cat("Dot Product:",dot(our_vector,C),"\n")
  cat("Magnitude:",Norm(C),"\n")
}
## Answer choice C is correct.

D. \(\lbrace 6,2,1 \rbrace\)

# install.packages("pracma")
library(pracma)
D <- c(6,2,1)
if (dot(our_vector,D) == 0 & Norm(D) == our_magnitude) {
  cat("Answer choice D is correct.")
} else {
  cat("Answer choice D is incorrect.","\n")
  cat("Dot Product:",dot(our_vector,D),"\n")
  cat("Magnitude:",Norm(D),"\n")
}
## Answer choice D is incorrect. 
## Dot Product: 0 
## Magnitude: 6.403124

We find that answer choice C is correct.

Question 2

The ages of domesticated cats are normally distributed with mean 15.7 years. A sample of twenty-five cats in a town were studied and the mean lifetime was found to be 16.5 years with a standard deviation of 1.6 years. If the lifetimes of domesticated cats follows a Normal distribution and the population mean is denoted by \(\mu\), test the following null and alternative hypotheses at the 1% level: \(H_0: \mu = 15.7\), \(H_1: \mu \neq 15.7\).

t.test(x = rnorm(n = 25,mean = 16.5,sd = 1.6),
       mu = 15.7,
       alternative = "two.sided",
       conf.level = 0.01)
## 
##  One Sample t-test
## 
## data:  rnorm(n = 25, mean = 16.5, sd = 1.6)
## t = -0.044829, df = 24, p-value = 0.9646
## alternative hypothesis: true mean is not equal to 15.7
## 1 percent confidence interval:
##  15.68120 15.68948
## sample estimates:
## mean of x 
##  15.68534

Question 3

Will, Yvette, and Zac are siblings.

Will says: “Five times my age minus two times Yvette’s age minus four times Zac’s age equals 27 years.”

Yvette says: “My age plus two times Will’s age minus three times Zac’s age equals 21 years.”

Zac says: “My age plus three times Will’s age minus three times Yvette’s age equals 30 years.”

How old is Will, Yvette, and Zac?

In order to solve this problem, we will need a few things. This includes variables and direct word-to-number translation for a system of equations.

Let \(W\) be Will’s age, \(Y\) be Yvette’s age, and \(Z\) be Zac’s age.

Will’s statement: \(5W - 2Y - 4Z = 27\) (direct translation of Will’s statement)

Yvette’s statement: \(Y + 2W - 3Z = 21\) (direct translation of Yvette’s statement)

Zac’s statement: \(Z + 3W - 3Y = 30\) (direct translation of Zac’s statement)

Summarize our system of equations…

\[5W - 2Y - 4Z = 27 \\ 2W + Y - 3Z = 21 \\ 3W - 3Y + Z = 30\]

I rearranged some of the terms to align with a data frame to get our answer.

q3_data <- data.frame(W = c(5,2,3), # W-coefficients
                      Y = c(-2,1,-3), # Y-coefficients
                      Z = c(-4,-3,1), # Z-coefficients
                      TOTALS = c(27,21,30)) # TOTALS: right-hand side of each equation
q3_model <- lm(TOTALS ~ . - 1,data = q3_data) # finding our coefficients
coef(q3_model) # extracting model coefficients
##  W  Y  Z 
## 21 15 12

We find that Will’s age is 21, Yvette’s age is 15, and Zac’s age is 12.

Question 4

What is \(\int e^x dx\)?

# install.packages(c("ggformula","mosaicCalc"))
library(ggformula)
## Warning: package 'ggformula' was built under R version 4.5.2
## Loading required package: ggplot2
## Loading required package: scales
## Loading required package: ggiraph
## Warning: package 'ggiraph' was built under R version 4.5.2
## Loading required package: ggridges
## Warning: package 'ggridges' was built under R version 4.5.2
## 
## New to ggformula?  Try the tutorials: 
##  learnr::run_tutorial("introduction", package = "ggformula")
##  learnr::run_tutorial("refining", package = "ggformula")
library(mosaicCalc)
## Warning: package 'mosaicCalc' was built under R version 4.5.2
## Registered S3 method overwritten by 'mosaic':
##   method                           from   
##   fortify.SpatialPolygonsDataFrame ggplot2
## 
## Attaching package: 'mosaicCalc'
## The following object is masked from 'package:stats':
## 
##     D
f <- makeFun(exp(x) ~ x)
anti_f <- antiD(f(x) ~ x)
anti_f
## function (x, C = 0) 
## exp(x) + C