Conceptual Questions

2.1 Scalars and Vectors

A weather forecast states the temperature is predicted to be $−5 ° C$ the following day. Is this temperature a vector or a scalar quantity? Explain.

Which of the following is a vector: a person’s height, the altitude on Mt. Everest, the velocity of a fly, the age of Earth, the boiling point of water, the cost of a book, Earth’s population, or the acceleration of gravity?

Give a specific example of a vector, stating its magnitude, units, and direction.

What do vectors and scalars have in common? How do they differ?

Suppose you add two vectors $\vec{A}$ and $\vec{B}$ . What relative direction between them produces the resultant with the greatest magnitude? What is the maximum magnitude? What relative direction between them produces the resultant with the smallest magnitude? What is the minimum magnitude?

Is it possible to add a scalar quantity to a vector quantity?

Is it possible for two vectors of different magnitudes to add to zero? Is it possible for three vectors of different magnitudes to add to zero? Explain.

Does the odometer in an automobile indicate a scalar or a vector quantity?

When a 10,000-m runner competing on a 400-m track crosses the finish line, what is the runner’s net displacement? Can this displacement be zero? Explain.

10.

A vector has zero magnitude. Is it necessary to specify its direction? Explain.

11.

Can a magnitude of a vector be negative?

12.

Can the magnitude of a particle’s displacement be greater that the distance traveled?

13.

If two vectors are equal, what can you say about their components? What can you say about their magnitudes? What can you say about their directions?

14.

If three vectors sum up to zero, what geometric condition do they satisfy?

2.2 Coordinate Systems and Components of a Vector

15.

Give an example of a nonzero vector that has a component of zero.

16.

Explain why a vector cannot have a component greater than its own magnitude.

17.

If two vectors are equal, what can you say about their components?

18.

If vectors $\vec{A}$ and $\vec{B}$ are orthogonal, what is the component of $\vec{B}$ along the direction of $\vec{A}$ ? What is the component of $\vec{A}$ along the direction of $\vec{B}$ ?

19.

If one of the two components of a vector is not zero, can the magnitude of the other vector component of this vector be zero?

20.

If two vectors have the same magnitude, do their components have to be the same?

2.4 Products of Vectors

21.

What is wrong with the following expressions? How can you correct them? (a) $C = \vec{A} \vec{B}$ , (b) $\vec{C} = \vec{A} \vec{B}$ , (c) $C = \vec{A} \times \vec{B}$ , (d) $C = A \vec{B}$ , (e) $C + 2 \vec{A} = B$ , (f) $\vec{C} = A \times \vec{B}$ , (g) $\vec{A} \cdot \vec{B} = \vec{A} \times \vec{B}$ , (h) $\vec{C} = 2 \vec{A} \cdot \vec{B}$ , (i) $C = \vec{A} / \vec{B}$ , and (j) $C = \vec{A} / B$ .

22.

If the cross product of two vectors vanishes, what can you say about their directions?

23.

If the dot product of two vectors vanishes, what can you say about their directions?

24.

What is the dot product of a vector with the cross product that this vector has with another vector?