Q: Given $$\mathbf{a} = \langle 1, 2, -1 \rangle$$ and $$\mathbf{b} = \langle 2, -1, 2 \rangle$$, what is the angle $$\theta$$ between them?

$$\theta = \arccos\left(\frac{-2}{\sqrt{6}\sqrt{9}}\right)$$

Question 1

Let $$\mathbf{v} = \langle 3, -4, 12 \rangle. $$What is the magnitude of $$\mathbf{v}$$?

Accepted Answer

$$\sqrt{3^2$ + (-4)^2$ + 12^2$} = 13$$

Question 2

Find a unit vector in the direction of $$\mathbf{u} = \langle -2, 1, 2 \rangle.$$

Accepted Answer

$$\left\langle -\frac{2}{3}, \frac{1}{3}, \frac{2}{3} \right\rangle$$

Question 3

Given $$\mathbf{a} = \langle 1, 2, -1 angle$$ and $$\mathbf{b} = \langle 2, -1, 2 angle$$, what is the angle $$	heta$$ between them?

Accepted Answer

$$	heta = \arccos\left(\frac{-2}{\sqrt{6}\sqrt{9}}ight)$$

Question 4

Find the area of the parallelogram determined by $$\mathbf{a} = \langle 1, 0, 2 \rangle$$ and $$\mathbf{b} = \langle 3, -1, 1 \rangle.$$

Accepted Answer

$$\sqrt{18}$$

Question 5

Which of the following is the equation of the sphere centered at $$(1, -2, 3)$$ with radius $$5$$?

Accepted Answer

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

Question 6

Find the vector and parametric equations of the line through $$P(1, -2, 3)$$ and $$Q(4, 1, -1).$$

Accepted Answer

Vector: $$\mathbf{r}(t) = \langle 1, -2, 3 \rangle + t\langle 3, 3, -4 \rangle$$; Parametric: $$x = 1 + 3t,\ y = -2 + 3t,\ z = 3 - 4t$$

Question 7

Given the line $$x = 1 + 2t$$, $$y = 3 - t$$, $$z = 4 + 5t$$, what is its symmetric form?

Accepted Answer

$$\frac{x - 1}{2} = \frac{y - 3}{-1} = \frac{z - 4}{5}$$

Question 8

Find the equation of the plane passing through $$(1, 2, -1)$$ and normal to $$\mathbf{n} = \langle 2, -1, 3 \rangle.$$

Accepted Answer

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

Question 9

Determine whether the planes $$2x - y + z = 1$$ and $$4x - 2y + 2z = 7$$ are parallel, identical, or intersecting.

Accepted Answer

Parallel but not identical

Question 10

Find the distance from the point $$(1, -1, 2) to $$the plane $$2x - y + 2z = 5.$$

Accepted Answer

$$\frac{|2(1) - (-1) + 2(2) - 5|}{\sqrt{2^2$ + (-1)^2$ + 2^2$}} = 1$$

Question 11

Identify the surface $$x^2$$ + $$y^2$$ - $$z^2 = 1$.

Accepted Answer

A hyperboloid of one sheet

Question 12

Evaluate $$\lim_{t 	o 0} \left( \frac{\sin t}{t}, \frac{1 - \cos t}{t}, e^t ight).$$

Accepted Answer

$$(1, 0, 1)$$

Question 13

Find the arc length of $$\mathbf{r}(t) = \langle 3t, 4t, 12t \rangle$$ for $$0 \leq t \leq 2.$$

Accepted Answer

$$26$$

Question 14

A particle moves according to $$\mathbf{r}(t) = \langle t^2$, 2t, \ln(t + 1) \rangle. $$What is its velocity vector?

Accepted Answer

$$\langle 2t, 2, \frac{1}{t + 1} \rangle$$

Question 15

Given $$\mathbf{r}(t) = \langle t, t^2$, t^3$ \rangle at$$ $$t = 1$$, what is the tangential component $$a_T of $$the acceleration?

Accepted Answer

$$\frac{44}{\sqrt{14}}$$

Question 16

Let $$\mathbf{a} = \langle 1, 2, 3 angle$$ and $$\mathbf{b} = \langle 2, -1, 4 angle. $$What is $$\mathbf{a} 	imes \mathbf{b}$$?

Accepted Answer

$$\langle 11, 2, -5 \rangle$$

Question 17

Which of the following describes the traces of the surface $$z = x^2$ - y^2$ in the coordinate planes?

Accepted Answer

In the $$xz-$$plane ($$y=0$$): $$z = x^2$ (parabola); in the yz$-plane (x=0$): z$$ = $$-y^2 ($$downward parabola); in the $$xy-$$plane ($$z=0$$): $$x^2$$ = $$y^2 ($$lines $$y = x$$ and $$y = -x)$$

Question 18

A drone starts at $$(2, -1, 4)$$ and moves with constant velocity $$\mathbf{v} = \langle 3, 2, -1 \rangle. $$When does it reach altitude $$z = 0$$?

Accepted Answer

$$t = 4$$

Question 19

Let $$\mathbf{a}, \mathbf{b}, \mathbf{c} \in \mathbb{R}^3. $$What does it mean geometrically if $$\mathbf{a} \cdot (\mathbf{b} 	imes \mathbf{c}) = 0$$?

Accepted Answer

The vectors $$\mathbf{a}, \mathbf{b}, \mathbf{c}$$ are coplanar

Calculus III (Multivariable Calculus) Test 1 Review – Step-by-Step Study Guidance

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