Q: Let $$\vec{a} = (2, -1, 3)$$ and $$\vec{b} = (1, 4, -2). $$What is the dot product $$\vec{a} \cdot \vec{b}$$?

$$2 \times 1 + (-1) \times 4 + 3 \times (-2) = 2 - 4 - 6 = -8$$

Q: Suppose $$\vec{a} = (3, 0, 4)$$ and $$\vec{b} = (0, 5, 0). $$What is $$|\vec{a} \times \vec{b}|$$?

$$|\vec{a}| \cdot |\vec{b}| \cdot \sin \theta = 5 \times 5 \times 1 = 25$$

Q: If $$\vec{a} = (1,2,3)$$ and $$\vec{b} = (4,5,6)$$, what is $$\vec{a} \times \vec{b}$$?

$$(2 \times 6 - 3 \times 5, 3 \times 4 - 1 \times 6, 1 \times 5 - 2 \times 4) = (12 - 15, 12 - 6, 5 - 8) = (-3, 6, -3)$$

Q: Let $$\vec{a} = (1, 2, 2)$$ and $$\vec{b} = (2, 1, -2). $$What is the angle $$\theta$$ between $$\vec{a}$$ and $$\vec{b}$$?

$$\cos \theta = \dfrac{\vec{a} \cdot \vec{b}}{||\vec{a}||\,||\vec{b}||} = \dfrac{1 \times 2 + 2 \times 1 + 2 \times (-2)}{\sqrt{1^2$+2^2+2^2$}\,\sqrt{2^2$+1^2+(-2)^2$}} = \dfrac{2+2-4}{3\times 3} = 0$$

Question 1

Let $$\vec{a} = (2, -1, 3)$$ and $$\vec{b} = (1, 4, -2). $$What is the dot product $$\vec{a} \cdot \vec{b}$$?

Accepted Answer

$$2 	imes 1 + (-1) 	imes 4 + 3 	imes (-2) = 2 - 4 - 6 = -8$$

Question 2

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

Accepted Answer

The cross product is a vector orthogonal to both $$\vec{a}$$ and $$\vec{b}.$$

Question 3

Given the points $$P = (1, 2, 3)$$ and $$Q = (4, 6, 8)$$, what is the vector $$\overrightarrow{PQ}$$?

Accepted Answer

$$(4-1, 6-2, 8-3) = (3, 4, 5)$$

Question 4

A plane passes through the point $$(2, -1, 3)$$ and has normal vector $$\vec{n} = (1, 2, -2). $$Which of the following is the equation of the plane?

Accepted Answer

$$1(x-2) + 2(y+1) - 2(z-3) = 0$$

Question 5

Suppose $$\vec{a} = (3, 0, 4)$$ and $$\vec{b} = (0, 5, 0). $$What is $$|\vec{a} 	imes \vec{b}|$$?

Accepted Answer

$$|\vec{a}| \cdot |\vec{b}| \cdot \sin 	heta = 5 	imes 5 	imes 1 = 25$$

Question 6

A particle moves along the path $$\vec{r}(t) = (3\cos t, 3\sin t, 4t). $$What is the speed of the particle at time $$t$$?

Accepted Answer

$$\sqrt{(-3\sin t)^2$ + (3\cos t)^2$ + 4^2$} = \sqrt{9\sin^2$ t + 9\cos^2$ t + 16} = \sqrt{9 + 16} = 5$$

Question 7

Which of the following is the correct parameterization for the line passing through $$(1,2,3) in $$the direction of vector $$(4,-1,2)$$?

Accepted Answer

$$\vec{r}(t) = (1,2,3) + t(4,-1,2)$$

Question 8

If $$\vec{a} = (1,2,3)$$ and $$\vec{b} = (4,5,6)$$, what is $$\vec{a} 	imes \vec{b}$$?

Accepted Answer

$$(2 	imes 6 - 3 	imes 5, 3 	imes 4 - 1 	imes 6, 1 	imes 5 - 2 	imes 4) = (12 - 15, 12 - 6, 5 - 8) = (-3, 6, -3)$$

Question 9

A plane has equation $$2x - 3y + z = 7. $$What is a normal vector to this plane?

Accepted Answer

$$(2, -3, 1)$$

Question 10

Let $$\vec{a} = (1, 2, 2)$$ and $$\vec{b} = (2, 1, -2). $$What is the angle $$	heta$$ between $$\vec{a}$$ and $$\vec{b}$$?

Accepted Answer

$$\cos 	heta = \dfrac{\vec{a} \cdot \vec{b}}{||\vec{a}||\,||\vec{b}||} = \dfrac{1 	imes 2 + 2 	imes 1 + 2 	imes (-2)}{\sqrt{1^2$+2^2+2^2$}\,\sqrt{2^2$+1^2+(-2)^2$}} = \dfrac{2+2-4}{3	imes 3} = 0$$

Question 11

A drone flies along the path $$\vec{r}(t) = (t, t^2$, t^3$). $$What is the velocity vector at $$t=2$$?

Accepted Answer

$$(1, 2t, 3t^2$)$$ evaluated at $$t=2 is$$ $$(1, 4, 12)$$

Question 12

Which of the following is a unit vector in the direction of $$\vec{v} = (6, 8, 0)$$?

Accepted Answer

$$\left(\dfrac{6}{10}, \dfrac{8}{10}, 0\right) = (0.6, 0.8, 0)$$

Question 13

A force $$\vec{F} = (10, 0, 0) N is $$applied at the point $$(0, 2, 0) m. $$What is the torque about the origin?

Accepted Answer

$$(0, 2, 0) 	imes (10, 0, 0) = (0, 0, -20) Nm$$

Question 14

Which of the following statements is TRUE about the dot product?

Accepted Answer

If $$\vec{a} \cdot \vec{b} = 0$$, then $$\vec{a}$$ and $$\vec{b}$$ are orthogonal.

Question 15

A plane contains the points $$(1,2,3)$$, $$(2,0,1)$$, and $$(0,1,2). $$What is a normal vector to the plane?

Accepted Answer

Compute $$\vec{v}_1 = (2-1, 0-2, 1-3) = (1, -2, -2)$$ and $$\vec{v}_2 = (0-1, 1-2, 2-3) = (-1, -1, -1). $$Then $$\vec{v}_1 	imes \vec{v}_2 = ((-2)	imes(-1) - (-2)	imes(-1), -((1)	imes(-1) - (-2)	imes(-1)), (1)	imes(-1) - (-2)	imes(-1)) = (2-2, -( -1-2), -1-2) = (0, 3, -3)$$

Question 16

A curve is given by $$\vec{r}(t) = (t, t^2$, t^3$). $$What is the arclength from $$t=0 to$$ $$t=1$$?

Accepted Answer

$$\$$int_0$$^1 \sqrt{(1)^2$ + (2t)^2$ + (3t^2$)^2$$} \, dt = \$$int_0^1$$ \sqrt{1 + $$4t^2$$ + $$9t^4$$} \, dt$

Question 17

Which of the following is the correct formula for the projection of $$\vec{a}$$ onto $$\vec{b}$$?

Accepted Answer

$$\mathrm{proj}_{\vec{b}} \vec{a} = \dfrac{\vec{a} \cdot \vec{b}}{||\vec{b}||^2} \vec{b}$$

Question 18

A line has parametric equations $$x = 2 + 3t$$, $$y = -1 + 4t$$, $$z = 5 - t. $$Which point lies on the line?

Accepted Answer

$$(5, 3, 4)$$ when $$t=1$$

Question 19

A vector-valued function $$\vec{r}(t)$$ describes the position of a particle. Which of the following best describes the physical meaning of $$\vec{r}'(t)$$?

Accepted Answer

The velocity vector of the particle at time $$t.$$

Vectors and Vector-Valued Functions in Calculus: Study Notes

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