Q: If $$\vec{u} = (4, -2)$$ and $$\vec{v} = (-1, 3)$$, what is $$2\vec{u} - 3\vec{v}$$?

$$(2 \times 4 - 3 \times (-1), 2 \times (-2) - 3 \times 3) = (8 + 3, -4 - 9) = (11, -13)$$

Q: A force vector $$\vec{F}$$ acts at the origin and has components $$(5, 12). $$What is the direction (angle with the x-axis) of $$\vec{F}$$?

$$\theta = \arctan\left(\frac{12}{5}\right) \approx 67.4^\circ$$

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

$$(4 - 2 \times 1, 3 - 2 \times 2) = (2, -1)$$

Question 1

Which of the following best describes a vector?

Accepted Answer

A quantity that has both magnitude and direction, such as displacement.

Question 2

Given the vector $$\vec{v} = (3, -4, 12)$$, what is its magnitude?

Accepted Answer

$$\sqrt{3^2$ + (-4)^2$ + 12^2$} = \sqrt{9 + 16 + 144} = \sqrt{169} = 13$$

Question 3

If $$\vec{a} = (2, 5)$$ and $$\vec{b} = (1, -1)$$, what is $$\vec{a} + \vec{b}$$?

Accepted Answer

$$(3, 4)$$

Question 4

Which property of vector addition is illustrated by $$\vec{a} + \vec{b} = \vec{b} + \vec{a}$$?

Accepted Answer

Commutative property

Question 5

A plane travels 300 km east and then 400 km north. What is the magnitude of its displacement vector?

Accepted Answer

$$\sqrt{300^2$ + 400^2$} = \sqrt{90000 + 160000} = \sqrt{250000} = 500 km$$

Question 6

If $$\vec{u} = (4, -2)$$ and $$\vec{v} = (-1, 3)$$, what is $$2\vec{u} - 3\vec{v}$$?

Accepted Answer

$$(2 	imes 4 - 3 	imes (-1), 2 	imes (-2) - 3 	imes 3) = (8 + 3, -4 - 9) = (11, -13)$$

Question 7

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

Accepted Answer

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

Question 8

If $$\vec{p} = (1, 0, 3)$$ and $$\vec{q} = (0, 1, 0)$$, express $$2\vec{p} - \vec{q} in $$terms of the standard basis vectors $$\vec{i}, \vec{j}, \vec{k}.$$

Accepted Answer

$$2\vec{p} - \vec{q} = 2(1, 0, 3) - (0, 1, 0) = (2, -1, 6) = 2\vec{i} - \vec{j} + 6\vec{k}$$

Question 9

A force vector $$\vec{F}$$ acts at the origin and has components $$(5, 12). $$What is the direction (angle with the x-axis) of $$\vec{F}$$?

Accepted Answer

$$	heta = \arctan\left(\frac{12}{5}ight) \approx 67.4^\circ$$

Question 10

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

Accepted Answer

$$\vec{a} + \vec{b} = \vec{a} \cdot \vec{b}$$

Question 11

A car moves from point $$A(2, 5) to $$point $$B(7, 9). $$What is the displacement vector $$\vec{AB}$$?

Accepted Answer

$$(7 - 2, 9 - 5) = (5, 4)$$

Question 12

Which of the following vectors is a linear combination of the standard basis vectors $$\vec{i} = (1, 0, 0)$$, $$\vec{j} = (0, 1, 0)$$, and $$\vec{k} = (0, 0, 1)$$?

Accepted Answer

$$(2, -1, 3) = 2\vec{i} - \vec{j} + 3\vec{k}$$

Question 13

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

Accepted Answer

$$(4 - 2 	imes 1, 3 - 2 	imes 2) = (2, -1)$$

Question 14

Which of the following statements about vectors is TRUE?

Accepted Answer

The sum of two vectors is a vector.

Question 15

A drone flies from $$(0, 0, 0) to$$ $$(2, 3, 5). $$What is the length of its displacement vector?

Accepted Answer

$$\sqrt{2^2$ + 3^2$ + 5^2$} = \sqrt{4 + 9 + 25} = \sqrt{38} \approx 6.16$$

Question 16

If $$\vec{u} = (1, 2)$$ and $$\vec{v} = (3, 4)$$, which vector represents $$\vec{u} + \vec{v}$$?

Accepted Answer

$$(1 + 3, 2 + 4) = (4, 6)$$

Question 17

Which of the following is the correct definition of the Triangle Law for vector addition?

Accepted Answer

If two vectors are positioned so that the initial point of one is at the terminal point of the other, their sum is the vector from the initial point of the first to the terminal point of the second.

Question 18

If $$\vec{a} = (0, 3)$$ and $$\vec{b} = (2, 1)$$, what is $$\vec{a} + 2\vec{b}$$?

Accepted Answer

$$(0 + 2 	imes 2, 3 + 2 	imes 1) = (4, 5)$$

Question 19

Which of the following is the correct formula for the length of a vector $$\vec{v} = (v_1$, v_2$, v_3$)$$?

Accepted Answer

$$|\vec{v}| = \sqrt{v_1$^2 + v_2$^2 + v_3$^2}$$

Section 2.2 - Vectors

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