If necessary, review:

- Taste of Trigonometry Help
- Vectors in the Cartesian Plane Help
- 3D Coordinates Help
- Vectors in 3D Space Help
- Planes in 3D Space Help
- Lines in 3D Space Help

**Vectors and Catesian Three-Space Practice Test**

**Directions:** A good score is eight correct.

1. The dot product (3,5,0) • (−4,−6,2) is equal to

(a) the scalar quantity −4

(b) the vector (−12,−30,0)

(c) the scalar quantity −42

(d) a vector perpendicular to the plane containing them both

2. What does the graph of the equation *y* = 3 look like in Cartesian three-space?

(a) A plane perpendicular to the *y* axis

(b) A plane parallel to the *xy* plane

(c) A line parallel to the *y* axis

(d) A line parallel to the *xy* plane

3. Suppose vector **d** in the Cartesian plane begins at exactly (1,1) and ends at exactly (4,0). What is dir **d** , expressed to the nearest degree?

(a) 342°

(b) 18°

(c) 0°

(d) 90°

4. Suppose a line is represented by the equation ( *x* − 3)/2 = ( *y* + 4)/5 = *z* − 1. Which of the following is a point on this line?

(a) (−3,4,−1)

(b) (3,−4,1)

(c) (2,5,1)

(d) There is no way to determine such a point without more information

5. Let Δ *PQR* be a right triangle whose hypotenuse measures exactly 1 unit in length, and whose other two sides measure *p* meters and *q* meters, as shown in Fig. 9-14. Let *θ* be the angle whose apex is point *P* . Which of the following statements is true?

(a) sin *θ* = *p/q*

(b) sin *θ* = cos *φ*

(c) tan *φ* = tan *θ*

(d) cos *θ* = 1/(tan *φ* )

6. With reference to Fig. 9-14, the Pythagorean theorem can be used to demonstrate that

(a) sin *θ* + cos *φ* = 1

(b) tan *θ* − tan *φ* = 1

(c) (sin *θ* ) ^{2} + (cos *θ* ) ^{2} = 1

(d) (sin *θ* ) ^{2} − (cos *θ* ) ^{2} = 1

7. What is the cross product (2 **i** + 0 **j** + 0 **k** ) × (0 **i** + 2 **j** + 0 **k** )?

(a) 0 **i** + 0 **j** + 0 **k**

(b) 2 **i** + 2 **j** + 0 **k**

(c) 0 **i** + 0 **j** + 4 **k**

(d) The scalar 0

8. What is the sum of the two vectors (3,5) and (−5,−3)?

(a) (0,0)

(b) (8,8)

(c) (2,2)

(d) (−2,2)

9. If a straight line in Cartesian three-space has direction defined by **m** = 0 **i** + 0 **j** + 3 **k** , we can surmise

(a) that the line is parallel to the *x* axis

(b) that the line lies in the *yz* plane

(c) that the line lies in the *xy* plane

(d) none of the above

10. Suppose a plane passes through the origin, and a vector normal to the plane is represented by 4 **i** − 5 **j** + 8 **k** . The equation of this plane is

(a) 4 *x* − 5 *y* + 8 *z* = 0

(b) −4 *x* + 5 *y* − 8 *z* = 7

(c) ( *x* − 4) = ( *y* + 5) = ( *z* − 8)

(d) impossible to determine without more information

**Answers:**

1. c

2. a

3. a

4. b

5. b

6. c

7. c

8. d

9. d

10. a

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