Q: If $$\mathbf{a} = \langle 5, 12 \rangle$$, what is the magnitude of $$-2\mathbf{a}$$?

$$2 \times \sqrt{5^2$ + 12^2$} = 2 \times 13 = 26$$

Question 1

Which of the following numbers is NOT a rational number?

Accepted Answer

$$\sqrt{2}$$

Question 2

A point $$P is $$located at $$(-5, 4) in $$the Cartesian coordinate system. In which quadrant does $$P$$ lie?

Accepted Answer

Quadrant II

Question 3

Given the vector $$\mathbf{a} = \langle 3, -4 \rangle$$, what is its magnitude?

Accepted Answer

$$\sqrt{3^2$ + (-4)^2$} = 5$$

Question 4

If a vector $$\mathbf{b} = \langle -2, 5 \rangle is $$multiplied by the scalar $$-3$$, what is the resulting vector?

Accepted Answer

$$\langle 6, -15 \rangle$$

Question 5

Which of the following statements about the zero vector $$\mathbf{0} = \langle 0, 0 \rangle is $$TRUE?

Accepted Answer

It has no specific direction.

Question 6

Given two vectors $$\mathbf{a} = \langle 1, 2 \rangle$$ and $$\mathbf{b} = \langle 3, -1 \rangle$$, what is $$\mathbf{a} + \mathbf{b}$$?

Accepted Answer

$$\langle 4, 1 \rangle$$

Question 7

A vector $$\mathbf{c} = \langle 8, 6 \rangle is $$normalized to form a unit vector $$\hat{c}. $$What is $$\hat{c}$$?

Accepted Answer

$$\left\langle \frac{4}{5}, \frac{3}{5} \right\rangle$$

Question 8

If $$\mathbf{a} = \langle a_1$, a_2$ \rangle$$ and $$\mathbf{b} = \langle b_1$, b_2$ \rangle$$, which formula correctly gives the difference $$\mathbf{a} - \mathbf{b}$$?

Accepted Answer

$$\langle a_1$ - b_1$, a_2$ - b_2$ \rangle$$

Question 9

A drone travels from point $$A(2, 3) to $$point $$B(7, 11). $$What is the vector representing this displacement?

Accepted Answer

$$\langle 5, 8 \rangle$$

Question 10

Which of the following best describes the geometric meaning of the parallelogram law of vector addition?

Accepted Answer

The sum of two vectors is represented by the diagonal of the parallelogram formed by the vectors as adjacent sides.

Question 11

If $$\mathbf{a} = \langle 4, 2 \rangle$$ and $$\mathbf{b} = \langle 1, 5 \rangle$$, what is the magnitude of $$\mathbf{a} + \mathbf{b}$$?

Accepted Answer

$$\sqrt{(4+1)^2$ + (2+5)^2$} = \sqrt{25 + 49} = \sqrt{74}$$

Question 12

A rescue boat moves $$3 km $$east and then $$4 km $$north. What is the magnitude of its displacement from the starting point?

Accepted Answer

$$\sqrt{3^2$ + 4^2$} = 5 km$$

Question 13

Which of the following is a unit vector in the direction of $$\mathbf{a} = \langle 6, 8 \rangle$$?

Accepted Answer

$$\left\langle \frac{3}{5}, \frac{4}{5} \right\rangle$$

Question 14

If $$\mathbf{a} = \langle 2, -1 \rangle$$ and $$\mathbf{b} = \langle -3, 4 \rangle$$, what is $$2\mathbf{a} - \mathbf{b}$$?

Accepted Answer

$$\langle 7, -6 \rangle$$

Question 15

Which of the following statements about basis vectors $$\mathbf{i} = \langle 1, 0 \rangle$$ and $$\mathbf{j} = \langle 0, 1 \rangle is $$correct?

Accepted Answer

Any vector in $$\mathbb{R}^2$$ can be written as a linear combination of $$\mathbf{i}$$ and $$\mathbf{j}.$$

Question 16

A car travels from $$A(1, 2) to$$ $$B(4, 6)$$, then from $$B to$$ $$C(7, 3). $$What is the total displacement vector from $$A to$$ $$C$$?

Accepted Answer

$$\langle 6, 1 \rangle$$

Question 17

If $$\mathbf{a} = \langle 5, 12 \rangle$$, what is the magnitude of $$-2\mathbf{a}$$?

Accepted Answer

$$2 	imes \sqrt{5^2$ + 12^2$} = 2 	imes 13 = 26$$

Question 18

Which of the following is an example of an irrational number?

Accepted Answer

$$\pi$$

Question 19

If $$\mathbf{a} = \langle 2, 3 \rangle$$ and $$\mathbf{b} = \langle 2, 3 \rangle$$, which of the following is TRUE?

Accepted Answer

$$\mathbf{a}$$ and $$\mathbf{b}$$ are equal vectors.

Foundations of Vectors and the Cartesian Plane: Essential Pre-Calculus Concepts

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