Review the following concepts if needed:

**Three-Space and Vectors Practice Test**

A good score is eight correct.

1. The magnitude (or length) of the cross product of two vectors depends on

(a) the angle between them

(b) their magnitudes

(c) their magnitudes and the angle between them

(d) neither their magnitudes nor the angle between them

2. The dot product of two vectors depends on

(a) the angle between them

(b) their magnitudes

(c) their magnitudes and the angle between them

(d) neither their magnitudes nor the angle between them

3. In spherical coordinates, the position of a point is specified by

(a) two angles and a distance

(b) two distances and an angle

(c) three distances

(d) three angles

4. Suppose you see a balloon hovering in the sky over a calm ocean. You are told that it is at azimuth 30°, that it is 3500 meters above the ocean surface, and that the point directly underneath it is 5000 meters away from you. This information is an example of the position of the balloon expressed in a form of

(a) Cartesian coordinates

(b) cylindrical coordinates

(c) spherical coordinates

(d) celestial coordinates

5. Suppose vector **a** , represented in Cartesian three-space by (3,–1,–5), is multiplied by a constant *k* = 2. What represents the product *k* a?

(a) The ordered triple (6,–2,–10)

(b) The ordered triple (–6,2,10)

(c) The scalar –6

(d) Nothing; *k a* is not defined because a vector cannot be multiplied by a scalar

6. Suppose there are two vectors that correspond to the Cartesian ordered pairs **a** = (3,5) and **b** = (1,0). What is the dot product of these vectors?

(a) (3,0)

(b) (4,5)

(c) (0,0)

(d) None of the above

7. Two vectors point in opposite directions, and one has twice the magnitude of the other. Their cross product

(a) points in the direction of the vector with the larger magnitude

(b) points in the direction of the vector with the smaller magnitude

(c) points in a direction perpendicular to the plane containing both vectors

(d) has zero magnitude

8. A vector denotes a phenomenon that has

(a) an abscissa and an ordinate

(b) a radius and an angle

(c) a magnitude and a direction

(d) an azimuth and an elevation

9. In Cartesian three-space,

(a) the axes are all mutually perpendicular

(b) *θ* represents azimuth, ø represents elevation, and *r* represents radius

(c) vectors are represented by an angle and a radius

(d) coordinates are all defined by angles

10. Refer to Fig. 6-8 in the chapter text. What is the dot product of the vectors ( *θ,r* ) and ( *θ,kr* )?

(a) *–kr* ^{2}

(b) *kr* ^{2}

(c) *k* ^{2} *r* ^{2}

(d) It is impossible to determine this without more information

**Answers:**

1. c

2. c

3. a

4. b

5. a

6. d

7. d

8. c

9. a

10. b

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