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1. Given the vectors:

$\vec{v} = \left( 1,0,7 \right)$

$\vec{w} = \left( 0,-1,2 \right)$

find the distance between them, $d \left( \vec{v}, \vec{w} \right)$

  • $-2$
  • $\sqrt{ \left( 27 \right) }$
  • $5$
  • $\sqrt{ \left( 23 \right) }$

2. You are given the points $P: \left( 1, 0, -3 \right)$ and $Q: \left( −1, 0, -3 \right)$. The magnitude of the vector from $P$ to $Q$ is:

  • $-2$
  • $2$
  • $3$

3. Select the correct statements pertaining to the dot product.

  • The dot product vector is the diagonal in a parallelogram formed by the two vectors $\vec{u}$ and $\vec{v}$.
  • The dot product of two vectors is always a scalar.
  • The dot product of orthogonal vectors is always 0.
  • The dot product of orthogonal vectors is always 1.

4. Calculate the norm $|| v ||$ of the vector $\vec{v} = \left( 1,-5,2,0,-3 \right)$ and select the correct answer.

  • $|| v || = \sqrt{35}$
  • $|| v || = 5$
  • $|| v || = 39$
  • $|| v || = \sqrt{39}$

5. Which of the vectors has the greatest norm?

  • (A)

$$\begin{bmatrix} 0 \cr 0 \cr 0 \cr 0 \end{bmatrix}$$

  • (B)

$$\begin{bmatrix} 1 \cr 2 \cr -3 \end{bmatrix}$$

  • (C)

$$\begin{bmatrix} 2 \cr 5 \end{bmatrix}$$

  • (D)

$$\begin{bmatrix} 1 \cr 0 \cr -2 \cr 0 \cr -1 \end{bmatrix}$$

  • (E)

$$\begin{bmatrix} 2 \cr 2 \cr 2 \cr 2 \end{bmatrix}$$

6. Calculate the dot product $\vec{a} \cdot \vec{b}$ and select the correct answer.

$$\vec{a} = \begin{bmatrix} -1 \cr 5 \cr 2 \end{bmatrix}, \vec{b} = \begin{bmatrix} -3 \cr 6 \cr -4 \end{bmatrix}$$

  • (A)

$$\begin{bmatrix} 1 \cr 0 \cr 1 \end{bmatrix}$$

  • (B)

$$30$$

  • (C)

$$25$$

  • (D)

$$\begin{bmatrix} -3 \cr 30 \cr -8 \end{bmatrix}$$

7. Which of the following is the result of performing the multiplication $M_1 \cdot M_2$? Where $M_1$ and $M_2$ are given by:

$$M_1 = \begin{bmatrix} 2 & -1 \cr 3 & -3 \end{bmatrix}, M_2 = \begin{bmatrix} 5 & -2 \cr 0 & 1 \end{bmatrix}$$

  • (A)

$$\begin{bmatrix} 10 & 3 \cr 15 & 4 \end{bmatrix}$$

  • (B)

$$\begin{bmatrix} 10 & 15 \cr -3 & -4 \end{bmatrix}$$

  • (C)

$$\begin{bmatrix} 10 & -5 \cr 15 & -9 \end{bmatrix}$$

  • (D)

$$\begin{bmatrix} 10 & -3 & 1 \cr 15 & -4 & 0 \cr 1 & 0 & 1 \end{bmatrix}$$

8. Calculate the dot product $\vec{w} \cdot \vec{z}$ and select the correct answer.

$$\vec{w} = \begin{bmatrix} -9 \cr -1 \end{bmatrix}, \vec{z} = \begin{bmatrix} -3 \cr -5 \end{bmatrix}$$

  • (A)

$$32$$

  • (B)

$$\begin{bmatrix} -27 \cr -5 \end{bmatrix}$$

  • (C)

$$\begin{bmatrix} 27 \cr 5 \end{bmatrix}$$

  • (D)

$$35$$