Q: Which of the following best describes the process for finding the direction angle $$\theta of a $$vector $$\mathbf{v} = (a, b)$$?

$$\theta = \arctan\left(\frac{b}{a}\right)$$, adjusted for the correct quadrant

Q: A boat is heading north at $$20 km/h$$, but a current pushes it east at $$5 km/h. $$What is the boat's actual speed?

$$ ext{Actual speed} = \\sqrt{20^2$ + 5^2$} = 20.6 km/h$$

Question 1

Given the vector $$\mathbf{r} = (-4, -4)$$, which quadrant does it lie in?

Accepted Answer

Quadrant III

Question 2

Find the component form of the resultant vector $$\mathbf{u} + \mathbf{v}$$, where $$\mathbf{u} = (-2, 2)$$ and $$\mathbf{v} = (-5, -4).$$

Accepted Answer

$$(-7, -2)$$

Question 3

If $$\mathbf{f} = (-6, 3)$$ and $$\mathbf{g} = (-1, -3)$$, what is $$\mathbf{f} + \mathbf{g}$$?

Accepted Answer

$$(-7, 0)$$

Question 4

Find $$-4\mathbf{u} if$$ $$\mathbf{u} = (-9, 3).$$

Accepted Answer

$$(36, -12)$$

Question 5

If $$\mathbf{f} = (-12, 6)$$, what is $$8\mathbf{f}$$?

Accepted Answer

$$(-96, 48)$$

Question 6

Find the magnitude of the vector $$\mathbf{p} = (-4, 8).$$

Accepted Answer

$$\sqrt{80}$$

Question 7

Calculate the magnitude of $$\mathbf{v} = (-21, 28).$$

Accepted Answer

$$35$$

Question 8

A vector $$\mathbf{b} = (-9, -40) is $$given. What is its direction angle (measured counterclockwise from the positive x-axis)?

Accepted Answer

$$	heta = 203^	ext{o}$$

Question 9

For the vector $$\mathbf{m} = (-1, 10)$$, what is the direction angle (rounded to the nearest degree)?

Accepted Answer

$$	heta = 95^	ext{o}$$

Question 10

A twin-engine airplane flies directly south at $$300 mi/h in $$still air. It encounters a $$50 mi/h $$wind blowing due east. What is the magnitude of the resultant velocity?

Accepted Answer

$$	ext{Resultant speed} = 304 mi/h$$

Question 11

For the airplane in the previous question, what is the direction angle of the resultant velocity (measured east of south, rounded to the nearest degree)?

Accepted Answer

$$9^	ext{o}$$ 	ext{ east of south}

Question 12

Which of the following best describes the process for finding the direction angle $$	heta of a $$vector $$\mathbf{v} = (a, b)$$?

Accepted Answer

$$	heta = \arctan\left(\frac{b}{a}ight)$$, adjusted for the correct quadrant

Question 13

A vector $$\mathbf{v}$$ has magnitude $$10$$ and direction angle $$120^	ext{o}. $$What are its component forms $$(v_x, v_y)$$?

Accepted Answer

$$(-5, 8.66)$$

Question 14

If a vector $$\mathbf{a} = (x, y) is $$multiplied by a scalar $$k$$, what happens to its magnitude?

Accepted Answer

The magnitude becomes $$|k|$$ times the original magnitude.

Question 15

A vector $$\mathbf{w} = (a, b) is $$perpendicular to $$\mathbf{v} = (c, d) if$$:

Accepted Answer

$$ac + bd = 0$$

Question 16

A boat is heading north at $$20 km/h$$, but a current pushes it east at $$5 km/h. $$What is the boat's actual speed?

Accepted Answer

$$	ext{Actual speed} = \sqrt{20^2$ + 5^2$} = 20.6 km/h$$

Question 17

If $$\mathbf{v} = (x, y)$$, which formula gives its magnitude?

Accepted Answer

$$\sqrt{x^2$ + y^2$}$$

Question 18

Which of the following is NOT a property of vector addition?

Accepted Answer

The magnitude of $$\mathbf{u} + \mathbf{v} is $$always equal to the sum of the magnitudes of $$\mathbf{u}$$ and $$\mathbf{v}.$$

Trigonometry Vectors Worksheet – Step-by-Step Guidance

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