Q: Using the dot product, what is the angle $$\theta$$ between $$\vec{A} = 3\hat{i} - \hat{j}$$ and $$\vec{B} = \hat{i} + \hat{j}$$?

$$\theta = \arccos\left(\frac{2}{\sqrt{10} \cdot \sqrt{2}}\right)$$

Q: If a vector $$\vec{A}$$ has components $$A_x = 3$$, $$A_y = -1$$, what is the angle it makes with the positive x-axis?

$$\theta = \arctan\left(\frac{-1}{3}\right)$$

Question 1

Given the vectors $$\vec{A} = 3\hat{i} - \hat{j}$$ and $$\vec{B} = \hat{i} + \hat{j}$$, what is the magnitude of $$\vec{A}$$?

Accepted Answer

$$\sqrt{10}$$

Question 2

What is the direction (angle with respect to the positive x-axis) of $$\vec{B} = \hat{i} + \hat{j}$$?

Accepted Answer

$$45^\circ$$

Question 3

Using the dot product, what is the angle $$	heta$$ between $$\vec{A} = 3\hat{i} - \hat{j}$$ and $$\vec{B} = \hat{i} + \hat{j}$$?

Accepted Answer

$$	heta = \arccos\left(\frac{2}{\sqrt{10} \cdot \sqrt{2}}ight)$$

Question 4

If $$\vec{C} = \vec{A} + \vec{B}$$, what is $$\vec{C} in $$component form?

Accepted Answer

$$4\hat{i} + 0\hat{j}$$

Question 5

What is the cross product $$\vec{F} = \vec{A} 	imes \vec{B} in $$component form?

Accepted Answer

$$0\hat{i} + 0\hat{j} + 4\hat{k}$$

Question 6

Using the right-hand rule and the formula $$|\vec{A} 	imes \vec{B}| = AB \sin	heta$$, what is the magnitude of $$\vec{A} 	imes \vec{B}$$?

Accepted Answer

$$4$$

Question 7

Which of the following is a unit vector in the direction of $$\vec{B} = \hat{i} + \hat{j}$$?

Accepted Answer

$$\frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{j}$$

Question 8

Differentiate $$g(x) = (x^2$ + 1) \sin x$$ with respect to $$x.$$

Accepted Answer

$$2x \sin x + (x^2$ + 1) \cos x$$

Question 9

Differentiate $$h(x) = \frac{x^2$ + 3x + 1}{x - 1}$$ with respect to $$x.$$

Accepted Answer

$$\frac{(2x + 3)(x - 1) - (x^2$ + 3x + 1)}{(x - 1)^2$}$$

Question 10

Differentiate $$q(x) = \cos(4x)$$ with respect to $$x.$$

Accepted Answer

$$-4 \sin(4x)$$

Question 11

A particle has position vector $$\vec{r}(t) = t^2$ \hat{i} + (3t - 1) \hat{j} + (2 - t^3$) \hat{k}. $$What is its velocity vector $$\vec{v}(t)$$?

Accepted Answer

$$2t \hat{i} + 3 \hat{j} - 3t^2$ \hat{k}$$

Question 12

What is the speed $$|\vec{v}(t)| of $$the particle at $$t = 1$$?

Accepted Answer

$$\sqrt{4 + 9 + 9} = \sqrt{22}$$

Question 13

What is the acceleration vector $$\vec{a}(t)$$ for the particle with position $$\vec{r}(t) = t^2$ \hat{i} + (3t - 1) \hat{j} + (2 - t^3$) \hat{k}$$?

Accepted Answer

$$2 \hat{i} - 6t \hat{k}$$

Question 14

For $$f(x) = x^3$ - 6x^2$ + 9x$$, what are the critical points?

Accepted Answer

$$x = 0$$ and $$x = 3$$

Question 15

For $$f(x) = x^3$ - 6x^2$ + 9x$$, which critical point is a minimum?

Accepted Answer

$$x = 3$$

Question 16

For $$f(x) = x^3$ - 6x^2$ + 9x$$, which critical point is a maximum?

Accepted Answer

$$x = 0$$

Question 17

A vector $$\vec{D} = \vec{A} - \vec{B}$$, where $$\vec{A} = 3\hat{i} - \hat{j}$$ and $$\vec{B} = \hat{i} + \hat{j}$$, is given. What is $$\vec{D} in $$component form?

Accepted Answer

$$2\hat{i} - 2\hat{j}$$

Question 18

If a vector $$\vec{A}$$ has components $$A_x = 3$$, $$A_y = -1$$, what is the angle it makes with the positive x-axis?

Accepted Answer

$$	heta = \arctan\left(\frac{-1}{3}ight)$$

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