Q: The dot product of $$\mathbf{a} = \langle 1, 2, 3 \rangle$$ and $$\mathbf{b} = \langle 4, -5, 6 \rangle is$$:

$$1 \times 4 + 2 \times (-5) + 3 \times 6 = 12$$

Q: Given the matrix $$A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$$, what is $$\det(A)$$?

$$1 \times 4 - 2 \times 3 = -2$$

Q: Given vectors $$\mathbf{a} = \langle 1, 2, 3 \rangle$$ and $$\mathbf{b} = \langle 4, 5, 6 \rangle$$, what is $$\mathbf{a} \cdot \mathbf{b}$$?

$$1 \times 4 + 2 \times 5 + 3 \times 6 = 32$$

Question 1

Which of the following statements about vectors is TRUE?

Accepted Answer

A vector can be represented as an ordered pair or triple of real numbers.

Question 2

Given vectors $$\mathbf{a} = \langle 2, -1 \rangle$$ and $$\mathbf{b} = \langle -3, 4 \rangle$$, what is $$\mathbf{a} + \mathbf{b}$$?

Accepted Answer

$$\langle -1, 3 \rangle$$

Question 3

If $$\mathbf{v} = \langle 3, 4 \rangle$$, what is the unit vector in the direction of $$\mathbf{v}$$?

Accepted Answer

$$\langle \frac{3}{5}, \frac{4}{5} \rangle$$

Question 4

The dot product of $$\mathbf{a} = \langle 1, 2, 3 \rangle$$ and $$\mathbf{b} = \langle 4, -5, 6 \rangle is$$:

Accepted Answer

$$1 	imes 4 + 2 	imes (-5) + 3 	imes 6 = 12$$

Question 5

Which of the following is the correct formula for the distance between points $$A(x_1$, y_1$, z_1$)$$ and $$B(x_2$, y_2$, z_2$) in$$ $$\mathbb{R}^3$$?

Accepted Answer

$$\sqrt{(x_2$ - x_1$)^2$$ + ($$y_2$$ - $$y_1)^2$ + (z_2$ - z_1$)^2$$}$

Question 6

A force $$\mathbf{F} of $$magnitude $$10 N $$acts at an angle of $$60^\circ to $$the horizontal. What is the $$i ($$horizontal) component of $$\mathbf{F}$$?

Accepted Answer

$$10 \cos 60^\circ$$

Question 7

Which of the following statements about the cross product $$\mathbf{a} 	imes \mathbf{b} is $$TRUE?

Accepted Answer

It is a vector perpendicular to both $$\mathbf{a}$$ and $$\mathbf{b}.$$

Question 8

Given the matrix $$A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$$, what is $$\det(A)$$?

Accepted Answer

$$1 	imes 4 - 2 	imes 3 = -2$$

Question 9

Which of the following systems of equations has no solution?

Accepted Answer

$$x + y = 2$$ and $$x + y = 3$$

Question 10

If $$z = 1 + i$$, what is $$|z|$$?

Accepted Answer

$$\sqrt{2}$$

Question 11

Which of the following is the modulus-argument (polar) form of $$z = -1 + i$$?

Accepted Answer

$$\sqrt{2}(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4})$$

Question 12

Suppose $$A is a$$ $$2 	imes 2$$ invertible matrix. Which of the following statements is TRUE?

Accepted Answer

The inverse $$A^{-1}$$ exists and is unique.

Question 13

Given vectors $$\mathbf{a} = \langle 1, 2, 3 \rangle$$ and $$\mathbf{b} = \langle 4, 5, 6 \rangle$$, what is $$\mathbf{a} \cdot \mathbf{b}$$?

Accepted Answer

$$1 	imes 4 + 2 	imes 5 + 3 	imes 6 = 32$$

Question 14

If $$z = 2e$$^{i\frac{\pi}$${3}}$$, what is $$z in $$standard form?

Accepted Answer

$$2(\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}) = 1 + i\sqrt{3}$$

Question 15

Which of the following is the correct formula for the area of the parallelogram defined by vectors $$\mathbf{a}$$ and $$\mathbf{b} in$$ $$\mathbb{R}^3$$?

Accepted Answer

$$|\mathbf{a} 	imes \mathbf{b}|$$

Question 16

If $$A is a$$ $$3 	imes 3$$ matrix with $$\det(A) = 0$$, which of the following is TRUE?

Accepted Answer

A has no inverse.

Question 17

Which of the following is the correct formula for the volume of the parallelepiped defined by vectors $$\mathbf{u}$$, $$\mathbf{v}$$, and $$\mathbf{w}$$?

Accepted Answer

$$|\mathbf{u} \cdot (\mathbf{v} 	imes \mathbf{w})|$$

Vectors and Three-Dimensional Geometry: Study Notes for Calculus Students

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