$$\vec{v} \times \vec{v} = \vec{0}$$

$$\begin{cases} x = -2 + 2t \\ y = 2 - t \\ z = 4 + 5t \end{cases}$$

Question 1

Which of the following statements about the dot product of two vectors $$\vec{u}$$ and $$\vec{v} in$$ $$\mathbb{R}^3 is $$always true?

Accepted Answer

$$\vec{u} \cdot \vec{v} = \vec{v} \cdot \vec{u}$$

Question 2

Given vectors $$\vec{a} = \vec{i} + \vec{j} - 2\vec{k}$$ and $$\vec{b} = 3\vec{i} - 2\vec{j} + \vec{k}$$, what is $$\vec{c} = \vec{a} + \vec{b}$$?

Accepted Answer

$$4\vec{i} - \vec{j} - \vec{k}$$

Question 3

Which of the following is the correct formula for the magnitude (length) of a vector $$\vec{v} = \langle v_1$, v_2$, v_3$ \rangle$$?

Accepted Answer

$$|\vec{v}| = \sqrt{v_1$^2 + v_2$^2 + v_3$^2}$$

Question 4

Which of the following statements about the cross product is correct?

Accepted Answer

$$\vec{v} 	imes \vec{v} = \vec{0}$$

Question 5

Which of the following is the general equation of a plane in $$\mathbb{R}^3$$?

Accepted Answer

$$Ax + By + Cz = D$$

Question 6

Given the plane $$2x - y + 5z = 12$$, what is a direction vector for a line perpendicular to this plane?

Accepted Answer

$$\langle 2, -1, 5 \rangle$$

Question 7

Which of the following is the correct parametric equation for the line through $$(-2, 2, 4)$$ perpendicular to the plane $$2x - y + 5z = 12$$?

Accepted Answer

$$\begin{cases} x = -2 + 2t \ y = 2 - t \ z = 4 + 5t \end{cases}$$

Question 8

Which of the following is the equation for a hyperboloid of two sheets with its axis along the $$y-$$axis?

Accepted Answer

$$-\frac{y^2$}{b^2$} + \frac{x^2$}{a^2$} + \frac{z^2$}{c^2$} = 1$$

Question 9

Which of the following is a unit vector?

Accepted Answer

$$\frac{1}{\sqrt{3}}\langle 1, 1, 1 \rangle$$

Question 10

If $$\vec{u} \cdot \vec{v} = 0$$ and neither $$\vec{u}$$ nor $$\vec{v} is $$the zero vector, what can be concluded?

Accepted Answer

$$\vec{u}$$ and $$\vec{v}$$ are orthogonal

Question 11

Which of the following is the correct formula for the orthogonal projection of $$\vec{u}$$ onto $$\vec{v}$$?

Accepted Answer

$$\mathrm{proj}_{\vec{v}} \vec{u} = \frac{\vec{u} \cdot \vec{v}}{|\vec{v}|^2} \vec{v}$$

Question 12

Which of the following statements is true about the cross product of two unit vectors in $$\mathbb{R}^3$$?

Accepted Answer

The cross product can be a unit vector if and only if the two vectors are orthogonal

Question 13

Given points $$A(1, 0, 0)$$, $$B(2, 0, -1)$$, and $$C(1, 4, 3)$$, which of the following is an equation of the plane containing these points?

Accepted Answer

$$4x + 2y + z = 4$$

Question 14

Which of the following is NOT a property of the dot product?

Accepted Answer

$$\vec{u} \cdot (\vec{v} 	imes \vec{w}) = (\vec{u} \cdot \vec{v}) 	imes (\vec{u} \cdot \vec{w})$$

Question 15

Which of the following best describes the trace of a surface?

Accepted Answer

The intersection of the surface with a coordinate plane

Question 16

If two planes are parallel, which of the following must be true?

Accepted Answer

Their normal vectors are parallel

Question 17

Which of the following is the distance from point $$P(x_0$, y_0$, z_0$) to $$the line $$\vec{r}(t) = \vec{r}_0 + t\vec{v}$$?

Accepted Answer

$$\frac{|(\vec{r}_0 - \vec{p}) 	imes \vec{v}|}{|\vec{v}|}$$

Question 18

Which of the following is a coordinate unit vector in $$\mathbb{R}^3$$?

Accepted Answer

$$\vec{i} = \langle 1, 0, 0 \rangle$$

Multivariable Calculus Exam 1 Study Guide: Step-by-Step Guidance

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