Question 1

Which of the following best describes a scalar quantity?

Accepted Answer

A quantity that consists of a single real number specifying its magnitude.

Question 2

The gravitational force $$\vec{F}$$ exerted by the earth on an artificial satellite is a vector quantity. If $$\vec{F} is $$directed towards the center of the earth and its magnitude is proportional to $$\frac{1}{r^2$}$$, where $$r is $$the distance from the center of the earth to the satellite, which of the following statements is true?

Accepted Answer

$$\vec{F}$$ has both magnitude and direction, and its direction is always towards the center of the earth.

Question 3

Given a vector $$\vec{v}$$, what is the length (or magnitude) of $$\vec{v} in $$terms of its components $$v_1$$ and $$v_2$$?

Accepted Answer

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

Question 4

If $$\vec{u} = \langle 3, 4 \rangle$$, what is the magnitude of $$\vec{u}$$?

Accepted Answer

$$5$$

Question 5

Which of the following is true about vector addition?

Accepted Answer

$$\vec{u} + \vec{v} = \vec{v} + \vec{u}$$

Question 6

Given $$\vec{u} = \langle 2, -5 \rangle$$ and $$\vec{v} = \langle 3, 4 \rangle$$, compute $$2\vec{u} - 3\vec{v}.$$

Accepted Answer

$$\langle -5, -23 \rangle$$

Question 7

Which of the following is the correct definition of a unit vector in the direction of $$\vec{v}$$?

Accepted Answer

$$\vec{u} = \frac{\vec{v}}{|\vec{v}|}$$

Question 8

If $$c is a $$scalar and $$\vec{v} is a $$vector, what is the result of $$c\vec{v}$$ when $$c \u003c 0$$?

Accepted Answer

A vector in the opposite direction to $$\vec{v}$$, scaled by $$|c|.$$

Question 9

Which of the following is the zero vector $$\vec{0}$$?

Accepted Answer

A vector with zero length and no direction.

Question 10

If $$\vec{v} = \langle v_1$, v_2$ \rangle$$ and $$\vec{w} = \langle w_1$, w_2$ \rangle$$, what is $$\vec{v} + \vec{w} in $$coordinate form?

Accepted Answer

$$\langle v_1$ + w_1$, v_2$ + w_2$ \rangle$$

Question 11

A river flows with velocity $$\vec{v} = \langle 2, 3 \rangle m/s. $$What is the speed of the water?

Accepted Answer

$$\sqrt{13} m/s$$

Question 12

Which of the following is true about the position vector $$\vec{v} in $$the coordinate plane?

Accepted Answer

It is defined by the coordinates of its head, $$\langle v_1$, v_2$ \rangle.$$

Question 13

Given $$\vec{v} = \langle 6, 8 \rangle$$, find a unit vector in the direction of $$\vec{v}.$$

Accepted Answer

$$\langle 0.6, 0.8 \rangle$$

Question 14

If $$\vec{u} = \langle 1, 2 \rangle$$ and $$\vec{v} = \langle 3, -1 \rangle$$, what is $$\vec{u} - \vec{v}$$?

Accepted Answer

$$\langle -2, 3 \rangle$$

Question 15

Which of the following is a property of the zero vector $$\vec{0}$$?

Accepted Answer

For any vector $$\vec{v}$$, $$\vec{v} + \vec{0} = \vec{v}$$

Question 16

If $$\vec{v} = \langle 2, 1 \rangle$$, find a vector of length $$3 in $$the same direction as $$\vec{v}.$$

Accepted Answer

$$\langle \frac{6}{\sqrt{5}}, \frac{3}{\sqrt{5}} \rangle$$

Question 17

Which of the following statements is true about two vectors $$\vec{u}$$ and $$\vec{v}$$?

Accepted Answer

They are equal if they have the same length and direction, regardless of their initial points.

Question 18

If $$\vec{v} = \langle v_1$, v_2$ \rangle$$, what is $$-\vec{v}$$?

Accepted Answer

$$\langle -v_1$, -v_2$ \rangle$$

Question 19

A vector $$\vec{v} is $$said to be parallel to the zero vector $$\vec{0}$$ because:

Accepted Answer

The zero vector has no direction, so every vector is considered parallel to it.

Vectors: Definitions, Properties, and Operations

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