Q: What is the standard equation of a conic section in polar coordinates with eccentricity $$e$$ and directrix at $$r = d$$?

$$r = \frac{ed}{1 - e \cos \theta}$$

Q: If $$\vec{u} = \langle 2, 1, 6 \rangle$$ and $$\vec{v} = \langle 1, 1, 1 \rangle$$, what is their dot product?

$$2 \times 1 + 1 \times 1 + 6 \times 1 = 9$$

Q: If $$\vec{u} = \langle 1, 2, 3 \rangle$$ and $$\vec{v} = \langle 4, -5, 6 \rangle$$, what is $$\vec{u} \cdot \vec{v}$$?

$$1 \times 4 + 2 \times (-5) + 3 \times 6 = 4 - 10 + 18 = 12$$

Question 1

Which of the following best describes a hyperbola in terms of its geometric definition?

Accepted Answer

It is the set of all points (x, y) such that the absolute value of the difference of the distances from two fixed points (the foci) is constant.

Question 2

Given the equation of a hyperbola $$\frac{x^2$}{a^2$} - \frac{y^2$}{b^2$} = 1$$, what is the relationship between $$a$$, $$b$$, and $$c ($$where $$c is $$the distance from the center to a focus)?

Accepted Answer

$$c^2$$ = $$a^2$$ + $$b^2$$

Question 3

What is the standard equation of a conic section in polar coordinates with eccentricity $$e$$ and directrix at $$r = d$$?

Accepted Answer

$$r = \frac{ed}{1 - e \cos 	heta}$$

Question 4

A vector $$\vec{v} in $$component form is $$\langle 3, -4 \rangle. $$What is its magnitude?

Accepted Answer

$$\sqrt{3^2$ + (-4)^2$} = 5$$

Question 5

If $$\vec{u} = \langle 2, 1, 6 \rangle$$ and $$\vec{v} = \langle 1, 1, 1 \rangle$$, what is their dot product?

Accepted Answer

$$2 	imes 1 + 1 	imes 1 + 6 	imes 1 = 9$$

Question 6

Which of the following statements is true about parallel vectors?

Accepted Answer

Two vectors are parallel if and only if one is a scalar multiple of the other.

Question 7

Given points $$P(6,7)$$ and $$Q(2,4)$$, what is the vector $$\overrightarrow{PQ} in $$component form?

Accepted Answer

$$\langle 2-6, 4-7 \rangle = \langle -4, -3 \rangle$$

Question 8

A sphere in $$\mathbb{R}^3$$ has center $$(1, -2, 3)$$ and radius $$5. $$What is its equation?

Accepted Answer

$$(x-1)^2$$ + $$(y+2)^2$$ + $$(z-3)^2 = 25$

Question 9

If $$\vec{u} = \langle 4, 0 \rangle$$ and $$\vec{v} = \langle 2, 0 \rangle$$, which statement is correct?

Accepted Answer

$$\vec{u}$$ and $$\vec{v}$$ are parallel.

Question 10

A plane flies due west at $$320 mi/h. $$There is a wind blowing $$40 mi/h $$toward the southwest ($$45^\circ$$ south of west). What is the magnitude of the plane's velocity relative to the ground?

Accepted Answer

$$\sqrt{(-320-20\sqrt{2})^2$ + (-20\sqrt{2})^2$} \approx 349.43 mi/h$$

Question 11

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

Accepted Answer

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

Question 12

Given $$\vec{u} = \langle 3, 4 \rangle$$, what is the unit vector in the direction of $$\vec{u}$$?

Accepted Answer

$$\langle 3/5, 4/5 \rangle$$

Question 13

If $$\vec{u} = \langle 1, 2, 3 \rangle$$ and $$\vec{v} = \langle 4, -5, 6 \rangle$$, what is $$\vec{u} \cdot \vec{v}$$?

Accepted Answer

$$1 	imes 4 + 2 	imes (-5) + 3 	imes 6 = 4 - 10 + 18 = 12$$

Question 14

Which of the following is true about the dot product of two orthogonal vectors?

Accepted Answer

The dot product is zero.

Question 15

Given the polar equation $$r = \frac{6}{1 - 0.5 \cos 	heta}$$, what type of conic section does this represent?

Accepted Answer

An ellipse

Question 16

If $$\vec{u} = \langle 2, -1 \rangle$$ and $$c = -3$$, what is $$c\vec{u}$$?

Accepted Answer

$$\langle -6, 3 \rangle$$

Vectors and Conic Sections: Study Notes for Calculus Students

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