8. Column Vectors

Question 1

LEVEL 6

Write the following expression as a single column vector.

\begin{pmatrix}1\\2\end{pmatrix} + \begin{pmatrix}-3\\5\end{pmatrix}

Select the correct answer from the list below:

A: \begin{pmatrix}-2\\7\end{pmatrix}

B: \begin{pmatrix}4\\7\end{pmatrix}

C: \begin{pmatrix}-3\\10\end{pmatrix}

D: \begin{pmatrix}2\\7\end{pmatrix}

CORRECT ANSWER:  A: \begin{pmatrix}-2\\7\end{pmatrix}

WORKED SOLUTION:

\begin{pmatrix}1\\2\end{pmatrix} + \begin{pmatrix}-3\\5\end{pmatrix} = \begin{pmatrix}1 - 3\\2+5\end{pmatrix} = \begin{pmatrix}-2\\7\end{pmatrix}

Question 2

LEVEL 6

Write the following expression as a single column vector.

Select the correct answer from the list below:

A:\begin{pmatrix}9\\-2\end{pmatrix}

B:\begin{pmatrix}14\\-4\end{pmatrix}

C:\begin{pmatrix}3\\-3\end{pmatrix}

D:\begin{pmatrix}9\\-3\end{pmatrix}

CORRECT ANSWER:  D:\begin{pmatrix}9\\-3\end{pmatrix}

WORKED SOLUTION:

2\begin{pmatrix}2\\-1\end{pmatrix} = \begin{pmatrix}2 \times 2\\2 \times -1\end{pmatrix} = \begin{pmatrix}4\\-2\end{pmatrix}

2\begin{pmatrix}2\\-1\end{pmatrix} + \begin{pmatrix}5\\-1\end{pmatrix} = \begin{pmatrix}4\\-2\end{pmatrix} + \begin{pmatrix}5\\-1\end{pmatrix} = \begin{pmatrix}4+5\\-2-1\end{pmatrix} = \begin{pmatrix}9\\-3\end{pmatrix}

Question 3

LEVEL 6

Given the following column vectors.

\textbf{a} = \begin{pmatrix}2\\-1\end{pmatrix},   \textbf{b} = \begin{pmatrix}-1\\3\end{pmatrix},  \textbf{c} = \begin{pmatrix}5\\4\end{pmatrix}

Write the following expressions as a single column vector.

Select the correct answer from the list below:

A:\begin{pmatrix}1\\2\end{pmatrix}

B:\begin{pmatrix}3\\-1\end{pmatrix}

C:\begin{pmatrix}3\\-2\end{pmatrix}

D:\begin{pmatrix}1\\-1\end{pmatrix}

CORRECT ANSWER:  A:\begin{pmatrix}1\\2\end{pmatrix}

WORKED SOLUTION:

\textbf{a}+\textbf{b} =  \begin{pmatrix}2\\-1\end{pmatrix} + \begin{pmatrix}-1\\3\end{pmatrix} = \begin{pmatrix}2-1\\-1+3\end{pmatrix} = \begin{pmatrix}1\\2\end{pmatrix}

Question 4

LEVEL 6

Given the following column vectors.

\textbf{a} = \begin{pmatrix}2\\-1\end{pmatrix},   \textbf{b} = \begin{pmatrix}-1\\3\end{pmatrix},  \textbf{c} = \begin{pmatrix}5\\4\end{pmatrix}

Write the following expressions as a single column vector.

Select the correct answer from the list below:

A: \begin{pmatrix}4\\-2\end{pmatrix}

B: \begin{pmatrix}-5\\-6\end{pmatrix}

C: \begin{pmatrix}9\\2\end{pmatrix}

D: \begin{pmatrix}9\\3\end{pmatrix}

CORRECT ANSWER:  C: \begin{pmatrix}9\\2\end{pmatrix}

WORKED SOLUTION:

2\textbf{a} = 2\begin{pmatrix}2\\-1\end{pmatrix} = \begin{pmatrix}2 \times 2\\2 \times -1\end{pmatrix} = \begin{pmatrix}4\\-2\end{pmatrix}

2\textbf{a}+\textbf{c}= 2\begin{pmatrix}2\\-1\end{pmatrix} + \begin{pmatrix}5\\4\end{pmatrix} = \begin{pmatrix}4\\-2\end{pmatrix} + \begin{pmatrix}5\\4\end{pmatrix} = \begin{pmatrix}4+5\\-2+4\end{pmatrix} = \begin{pmatrix}9\\2\end{pmatrix}

Question 5

LEVEL 6

Let \textbf{a}=\begin{pmatrix}5\\-4\end{pmatrix} and \textbf{b}=\begin{pmatrix}3\\2\end{pmatrix}. Write \textbf{a}+3\textbf{b} as a column vector.

Select the correct answer from the list below:

A: \begin{pmatrix}8\\4\end{pmatrix}

B: \begin{pmatrix}14\\2\end{pmatrix}

C: \begin{pmatrix}14\\10\end{pmatrix}

D: \begin{pmatrix}7\\1\end{pmatrix}

CORRECT ANSWER:  B: \begin{pmatrix}14\\2\end{pmatrix}

WORKED SOLUTION:

3\textbf{b}=3\times\begin{pmatrix}3\\2\end{pmatrix}= \begin{pmatrix}3 \times 3\\3 \times 2\end{pmatrix}  = \begin{pmatrix}9\\6\end{pmatrix}

\textbf{a}+3\textbf{b}=\begin{pmatrix}5\\-4\end{pmatrix} +\begin{pmatrix}9\\6\end{pmatrix}= \begin{pmatrix}5 + 9\\ -4 + 6\end{pmatrix} = \begin{pmatrix}14\\2\end{pmatrix}

Question 6

LEVEL 6

Let \textbf{a}=\begin{pmatrix}2\\2\end{pmatrix} and \textbf{b}=\begin{pmatrix}3\\-1\end{pmatrix}. Write 2\textbf{a}-5\textbf{b} as a column vector.

Select the correct answer from the list below:

A: \begin{pmatrix}-8\\4\end{pmatrix}

B: \begin{pmatrix}-11\\9\end{pmatrix}

C: \begin{pmatrix}11\\9\end{pmatrix}

D: \begin{pmatrix}9\\-11\end{pmatrix}

CORRECT ANSWER:    B: \begin{pmatrix}-11\\9\end{pmatrix}

WORKED SOLUTION:

2\textbf{a}=2\times\begin{pmatrix}2\\2\end{pmatrix}= \begin{pmatrix}2 \times 2\\2 \times 2\end{pmatrix} = \begin{pmatrix}4\\4\end{pmatrix}

5\textbf{b}=5\times\begin{pmatrix}3\\-1\end{pmatrix}= \begin{pmatrix}5 \times 3\\5 \times -1\end{pmatrix} = \begin{pmatrix}15\\-5\end{pmatrix}

2\textbf{a} - 5\textbf{b}= \begin{pmatrix}4\\4\end{pmatrix}-\begin{pmatrix}15\\-5\end{pmatrix} = \begin{pmatrix}4 - 15\\ 4- (-5)\end{pmatrix} = \begin{pmatrix}-11\\9\end{pmatrix}