Quiz Chapter 02: Vectors – Introduction to Matrix Algebra

1. A set of equations

- - $4x_{1} + 7x_{2} + 11x_{3} = 13$
  - $17x_{1} + 39x_{2} + 23x_{3} = 31$
  - $13x_{1} + 67x_{2} + 59x_{3} = 37$

can also be written as

x_{1} \begin{bmatrix} 4\\17\\13\\ \end{bmatrix} + x_{2} \begin{bmatrix} 7\\39\\23\\ \end{bmatrix} + x_{3} \begin{bmatrix}11\\23\\59\\ \end{bmatrix} = \begin{bmatrix} 13\\31\\37\\ \end{bmatrix}

4 \begin{bmatrix}x_{1}\\x_{2}\\x_{3}\\ \end{bmatrix} + 39 \begin{bmatrix}x_{1}\\x_{2}\\x_{3}\\ \end{bmatrix} + 59 \begin{bmatrix}x_{1}\\x_{2}\\x_{3}\\ \end{bmatrix} = \begin{bmatrix}13\\31\\37\\ \end{bmatrix}

x_{1} \begin{bmatrix}4\\7\\11\\ \end{bmatrix} + x_{2} \begin{bmatrix}17\\39\\23\\ \end{bmatrix} + x_{3} \begin{bmatrix}13\\67\\59\\ \end{bmatrix} = \begin{bmatrix}13\\31\\37\\ \end{bmatrix}

x_{1} \begin{bmatrix}13\\17\\4\\ \end{bmatrix} + x_{2} \begin{bmatrix}67\\39\\7\\ \end{bmatrix} + x_{3} \begin{bmatrix}59\\23\\11\\ \end{bmatrix} = \begin{bmatrix}57\\13\\31\\ \end{bmatrix}

2. The magnitude of the vector, $V=\left( 5, \, -3, \, 2 \right)$ is

4

10

\sqrt{38}

\sqrt{20}

3. The rank of the vector $\vec{A} \, \begin{bmatrix} 2\\3\\7\\ \end{bmatrix}, \, \begin{bmatrix} 6\\9\\21\\ \end{bmatrix}, \, \begin{bmatrix} 3\\2\\7\\ \end{bmatrix}$ is

1

2

3

4

4. If $\vec{A}=\left(5, \, 2, \, 3 \right)$ and $\vec{B}=\left(6, \, -7, \, 3 \right)$ , then $4\vec{A} + 5\vec{B}$ is

\left(50, \, -5, \, 6 \right)

\left(50, \, -27, \, 27 \right)

\left( 11, \, -5, \, 6 \right)

\left( 20, \, 8, \, 12 \right)

5. The dot product of two vectors $\vec{A}$ and $\vec{B}$

- - $\vec{A} = 3i + 5j + 7k$
  - $\vec{B} = 11i + 13j + 17k$

is most nearly

14.80

33.00

56.00

217.0

6. The angle in degrees between two vectors $\vec{u}$ and $\vec{v}$

- - $\vec{u} = 3i + 5j + 7k$
  - $\vec{v} = 11i + 13j + 17k$

is most nearly

8.124

11.47

78.52

81.88