Question 1

Let $$\mathbf{u} = \langle 3, -1 \rangle$$ and $$\mathbf{v} = \langle -3, 4 \rangle. $$Find $$-\mathbf{u} + 2\mathbf{v}.$$

Accepted Answer

$$\langle -9, 9 \rangle$$

Question 2

Express the vector $$\overrightarrow{AB} in $$the form $$ai + bj if$$ $$A is $$the point $$(-1, -7)$$ and $$B is $$the point $$(4, -14).$$

Accepted Answer

$$\mathbf{v} = 5i - 7j$$

Question 3

Given $$\mathbf{u} = -2i - 7j$$ and $$\mathbf{v} = -4i - 21j$$, find $$|\mathbf{v} - \mathbf{u}|.$$

Accepted Answer

$$10\sqrt{2}$$

Question 4

Let $$\mathbf{u} = 6i + j$$, $$\mathbf{v} = i + j$$, and $$\mathbf{w} = i - j. $$Find scalars $$a$$ and $$b$$ such that $$\mathbf{u} = a\mathbf{v} + b\mathbf{w}.$$

Accepted Answer

$$a = 7/2,\ b = 5/2$$

Question 5

A force of magnitude 8 pounds pulling on a suitcase makes an angle of $$30^\circ$$ with the ground. Express the force in terms of its $$i$$ and $$j$$ components.

Accepted Answer

$$6.928i + 4.000j$$

Question 6

An audio speaker that weighs 50 pounds hangs from the ceiling of a restaurant from two cables as shown in the figure. The left cable makes a $$45^\circ$$ angle and the right cable a $$35^\circ$$ angle with the ceiling. To two decimal places, what is the tension in the two cables?

Accepted Answer

Tension in right cable: 35.90 lb; tension in left cable: 41.59 lb

Question 7

Find the distance between points $$P_1(1$$, -1, -2)$$ and P_2$(5, -6, -5).$$

Accepted Answer

$$5\sqrt{2}$$

Question 8

If $$\mathbf{v} = \langle -1, 6, 0 \rangle$$, find $$|\mathbf{v}|.$$

Accepted Answer

$$\sqrt{37}$$

Question 9

Find the position vector for the vector having initial point $$P = (-1, -3, 0)$$ and terminal point $$Q = (4, 3, -3).$$

Accepted Answer

$$\mathbf{v} = 5i + 6j - 3k$$

Question 10

Express the vector in the form $$ai + bj + ck$$: $$2\mathbf{u} - 6\mathbf{v} if$$ $$\mathbf{u} = (1, 1, 0)$$ and $$\mathbf{v} = (3, 0, 1).$$

Accepted Answer

$$\mathbf{v} = -16i + 2j - 6k$$

Question 11

Find an equation for the sphere with center $$(-8, 10, 0)$$ and radius $$5.$$

Accepted Answer

$$x^2$$ + $$y^2$$ + $$z^2 + 16x - 20y = 139$

Question 12

Give a geometric description of the set of points whose coordinates satisfy $$x^2$$ + $$y^2$$ + $$z^2$$ \u003e 1$$.

Accepted Answer

All points outside the sphere of radius 1

Question 13

Find the center and radius of the sphere: $$x^2$$ + $$y^2$$ + $$z^2 - 18x - 10y - 6z = -15$.

Accepted Answer

Center: $$(9, 5, 3)$$, Radius: $$10$$

Question 14

A plane passes through the points $$A(1,2,3)$$, $$B(4,0,1)$$, and $$C(2,1,5). $$Which of the following is a normal vector to the plane?

Accepted Answer

$$\langle 4, -2, -3 \rangle$$

Question 15

A drone flies from point $$A(2, 3, 1) to $$point $$B(5, 7, 4). $$What is the displacement vector?

Accepted Answer

$$\langle 3, 4, 3 \rangle$$

Question 16

A car travels 10 km east, then 10 km north. What is the magnitude of its displacement from the starting point?

Accepted Answer

$$10\sqrt{2}$$

Question 17

Which of the following is the correct formula for the distance between points $$A(x_1$, y_1$, z_1$)$$ and $$B(x_2$, y_2$, z_2$) in 3D $$space?

Accepted Answer

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

Question 18

If $$\mathbf{a} = 2i + 3j - k$$ and $$\mathbf{b} = -i + 4j + 2k$$, what is $$\mathbf{a} + \mathbf{b}$$?

Accepted Answer

$$i + 7j + k$$

Question 19

A vector $$\mathbf{v}$$ has magnitude $$10$$ and makes an angle of $$60^\circ$$ with the positive $$x-$$axis. What are its $$i$$ and $$j$$ components?

Accepted Answer

$$8.66i + 5j$$

Question 20

Which of the following best describes the set of all points $$(x, y, z)$$ such that $$x^2$$ + $$y^2 = 9$?

Accepted Answer

A cylinder of radius 3 along the $$z-$$axis

Vectors and Geometry in Calculus: Study Notes

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