Q: A $$10 kg $$mass is pulled up a $$20 m $$ramp inclined at $$30^{\circ} to $$the horizontal by a force parallel to the ramp. What is the work done against gravity?

$$980 \sin 30$$^{\circ}$$ \times 20 = 9800 J$$

Question 1

Given the vectors $$\mathbf{v} = 3\mathbf{i} - 2\mathbf{j} + \mathbf{k}$$ and $$\mathbf{w} = 5\mathbf{i} + 6\mathbf{j} + 2\mathbf{k}$$, what is the dot product $$\mathbf{v} \cdot \mathbf{w}$$?

Accepted Answer

$$29$$

Question 2

If $$\mathbf{u} = (-5, 7)$$, $$\mathbf{v} = (1, 6)$$, and $$\mathbf{w} = (-11, 2)$$, evaluate $$\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}).$$

Accepted Answer

$$106$$

Question 3

Find the angle in degrees between $$\mathbf{v} = \mathbf{i} + \mathbf{j}$$ and $$\mathbf{w} = \mathbf{i} - \mathbf{j} + \mathbf{k}. $$Round to one decimal place.

Accepted Answer

$$35.3$$

Question 4

Given $$\mathbf{v} = \mathbf{i} + \mathbf{j} + \mathbf{k}$$ and $$\mathbf{u} = 12\mathbf{i} + 3\mathbf{j} + 4\mathbf{k}$$, find $$\mathrm{proj}_{\mathbf{v}} \mathbf{u}.$$

Accepted Answer

$$\frac{19}{3}\mathbf{i} + \frac{19}{3}\mathbf{j} + \frac{19}{3}\mathbf{k}$$

Question 5

Let $$\mathbf{u} = 4\mathbf{i} + 5\mathbf{j}$$ and $$\mathbf{v} = \mathbf{i} + \mathbf{j}. $$Find $$\mathrm{scal}_{\mathbf{v}} \mathbf{u}.$$

Accepted Answer

$$\frac{9\sqrt{2}}{2}$$

Question 6

Which of the following vectors is orthogonal to $$20\mathbf{i} - 8\mathbf{j}$$?

Accepted Answer

$$-10\mathbf{i} - 25\mathbf{j}$$

Question 7

A bullet is fired with a muzzle velocity of $$1163 ft/sec $$from a gun aimed at an angle of $$31^{\circ}$$ above the horizontal. What is the vertical component of the velocity?

Accepted Answer

$$999.0 ft/sec$$

Question 8

How much work does it take to slide a box $$6$$ meters along the ground by pulling it with a $$158 N $$force at an angle of $$15^{\circ}$$ from the horizontal?

Accepted Answer

$$915.7$$ Joules

Question 9

Given $$\mathbf{v} = 3\mathbf{i} - 2\mathbf{j} - 4\mathbf{k}$$ and $$\mathbf{w} = -2\mathbf{i} + 4\mathbf{j} + 3\mathbf{k}$$, find $$\mathbf{v} 	imes (4\mathbf{w}).$$

Accepted Answer

$$40\mathbf{i} - 4\mathbf{j} + 32\mathbf{k}$$

Question 10

Find the triple scalar product $$[\mathbf{u}, \mathbf{v}, \mathbf{w}] = (\mathbf{u} 	imes \mathbf{v}) \cdot \mathbf{w}$$ for $$\mathbf{u} = \mathbf{i} + \mathbf{j} + \mathbf{k}$$, $$\mathbf{v} = 8\mathbf{i} + 5\mathbf{j} + 2\mathbf{k}$$, $$\mathbf{w} = 9\mathbf{i} - 6\mathbf{j} + 4\mathbf{k}.$$

Accepted Answer

$$-3$$

Question 11

Given $$\mathbf{v} = 2\mathbf{i} + 3\mathbf{j} - \mathbf{k}$$ and $$\mathbf{w} = 4\mathbf{i} - 2\mathbf{j} - \mathbf{k}$$, find a vector orthogonal to both $$\mathbf{v}$$ and $$\mathbf{w}.$$

Accepted Answer

$$-5\mathbf{i} + 6\mathbf{j} + 8\mathbf{k}$$

Question 12

Find the area of the parallelogram determined by the points $$P(7, -5, 5)$$, $$Q(2, -7, -3)$$, $$R(2, 5, -2)$$, and $$S(7, 3, -10).$$

Accepted Answer

$$\sqrt{148529}$$

Question 13

A lever $$\overrightarrow{PQ} is$$ $$9 in. $$long and a force $$\overrightarrow{F} of$$ $$5 lb is $$applied at $$Q at an $$angle of $$45^{\circ} to $$the lever. What is the magnitude of the torque at $$P in $$foot-pounds?

Accepted Answer

$$\frac{15}{8}\sqrt{2} ft-lb$$

Question 14

A force $$\mathbf{F} = 10\mathbf{i} + 6\mathbf{j} N $$moves an object from the origin to the point $$(3, 4). $$How much work is done by the force?

Accepted Answer

$$54 J$$

Question 15

Let $$\mathbf{a} = 2\mathbf{i} - 3\mathbf{j} + 4\mathbf{k}$$ and $$\mathbf{b} = -\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}. $$Find the projection of $$\mathbf{a}$$ onto $$\mathbf{b}.$$

Accepted Answer

$$\frac{6}{9}(-\mathbf{i} + 2\mathbf{j} + 2\mathbf{k})$$

Question 16

A plane is flying with an airspeed of $$200 km/h in $$the direction $$30^{\circ}$$ north of east. If there is a wind blowing $$50 km/h $$due north, what is the plane's actual speed relative to the ground?

Accepted Answer

$$236.6 km/h$$

Question 17

Given $$\mathbf{u} = 2\mathbf{i} + 3\mathbf{j}$$ and $$\mathbf{v} = -6\mathbf{i} + 4\mathbf{j}$$, what is the value of $$\mathbf{u} \cdot \mathbf{v}$$?

Accepted Answer

Question 18

If $$\mathbf{a} = 2\mathbf{i} + \mathbf{j}$$ and $$\mathbf{b} = \mathbf{i} - 2\mathbf{j}$$, what is the angle between $$\mathbf{a}$$ and $$\mathbf{b}$$?

Accepted Answer

$$135^{\circ}$$

Question 19

A $$10 kg $$mass is pulled up a $$20 m $$ramp inclined at $$30^{\circ} to $$the horizontal by a force parallel to the ramp. What is the work done against gravity?

Accepted Answer

$$980 \sin 30$$^{\circ}$$ 	imes 20 = 9800 J$$

Question 20

Given $$\mathbf{a} = 3\mathbf{i} + 4\mathbf{j}$$ and $$\mathbf{b} = 4\mathbf{i} - 3\mathbf{j}$$, which of the following statements is true?

Accepted Answer

$$\mathbf{a}$$ and $$\mathbf{b}$$ are orthogonal.

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