1. Dots and Cross Products of Vectors in Space
LESSON 8–5
Dots and Cross Products of Vectors in Space

2. Five-Minute Check (over Lesson 8-4) TEKS Then/Now New Vocabulary
Key Concept: Dot Product and Orthogonal Vectors in Space
Example 1: Find the Dot Product to Determine Orthogonal Vectors in Space
Example 2: Angle Between Two Vectors in Space
Key Concept: Cross Product of Vectors in Space
Example 3: Find the Cross Product of Two Vectors
Example 4: Real-World Example: Torque Using Cross Product
Example 5: Area of a Parallelogram in Space
Key Concept: Triple Scalar Product
Example 6: Volume of a Parallelepiped
Lesson Menu

3. Find the length and the midpoint of the segment with endpoints at (−2, 3, 4) and (6, 1, −5).
B ;
C ;
D ;
5–Minute Check 1

4. Locate and graph v = (–3, 4, 2). A. B. C. D.
5–Minute Check 2

5. Which of the following represents 3x – 5y + z if x = 2, –7, 1, y = –5, 0, 3, and z = –1, 6, –4?
A. –23, –15, 14
B. 6, –1, –6
C. 30, −15, −16
D. 30, −15, 14
5–Minute Check 3

6. Targeted TEKS
P.4(J) Represent the addition of vectors and
the multiplication of a vector by a scalar
geometrically and symbolically.
P.4(K) Apply vector addition and
multiplication of a vector by a scalar in
mathematical and real-world problems.
Mathematical Processes
P.1(E), P.1(B)

7. You found the dot product of two vectors in the plane. (Lesson 8-3)
Find dot products of and angles between vectors in space.
Find cross products of vectors in space and use cross products to find area and volume.
Then/Now

8. cross product
torque
parallelepiped
triple scalar product
Vocabulary

9. Key Concept 1

10. Since u • v = 0, u and v are orthogonal.
Find the Dot Product to Determine Orthogonal Vectors in Space
A. Find the dot product of u and v for u = –1, 6, –3 and v = 3, –1, –3. Then determine if u and v are orthogonal.
u • v = –1(3) + 6(–1) + (–3)(–3)
= –3 + (–6) + 9 or 0
Since u • v = 0, u and v are orthogonal.
Answer: 0; orthogonal
Example 1

11. Since u • v ≠ 0, u and v are not orthogonal.
Find the Dot Product to Determine Orthogonal Vectors in Space
B. Find the dot product of u and v for u = 2, 4, –6 and v = –3, 2, 4. Then determine if u and v are orthogonal.
u • v = 2(–3) + 4(2) + (–6)(4)
= – (–24) or –22
Since u • v ≠ 0, u and v are not orthogonal.
Answer: –22; not orthogonal
Example 1

12. Find the dot product of u = –4, 5, –1 and v = 3, –3, 1
Find the dot product of u = –4, 5, –1 and v = 3, –3, 1. Then determine if u and v are orthogonal.
A. – 28; orthogonal
B. – 28; not orthogonal
C. – 4; orthogonal
D. – 4; not orthogonal
Example 1

13. Angle between two vectors
Angle Between Two Vectors in Space
Find the angle θ between u = –4, –1, –3 and v = 7, 3, 4 to the nearest tenth of a degree.
Angle between two vectors
u = –4, –1, –3 and v = 7, 3, 4
Evaluate the dot product and magnitudes.
Example 2

14. The measure of the angle between u and v is about 168.6°.
Angle Between Two Vectors in Space
Simplify.
Solve for θ.
The measure of the angle between u and v is about 168.6°.
Answer: °
Example 2

15. Find the angle between u = –2, 3, –1 and v = –4, –3, 4 to the nearest tenth of a degree.
C. 82.8°
D °
Example 2

16. Key Concept 3

17. Determinant of a 3 × 3 matrix
Find the Cross Product of Two Vectors
Find the cross product of u = 6, –1, –2 and v = –1, –4, 2. Then show that u × v is orthogonal to both u and v.
u = 6, –1, –2 and v = –1, –4, 2
Determinant of a 3 × 3 matrix
Determinants of 2 × 2 matrices
Example 3

18. Simplify. Component form
Find the Cross Product of Two Vectors
Simplify.
Component form
To show that u × v is orthogonal to both u and v, find the dot product of u × v with u and u × v with v.
(u × v) • u
= –10, –10, –25 • 6, –1, –2
= –10(6) + (–10)(–1) + (–25)( –2)
= – or 0
Example 3

19. Because both dot products are zero, the vectors are orthogonal.
Find the Cross Product of Two Vectors
(u × v) • v
= –10, –10, –25 • –1, –4, 2
= –10(–1) + (–10)(–4) + (–25)( 2)
= (–50) or 0
Because both dot products are zero, the vectors are orthogonal.
Answer: –10, –10, –25; u × v • u = –10, –10, –25 • 6, –1, –2 = – = 0 u × v • v = –10, –10, –25 • –1, –4, 2 = – 50 = 0
Example 3

20. Find the cross product of u = 2, 3, –1 and v = –3, 1, 4.
A. u × v = 13, 5, 11
B. u × v = 13, –5, 11
C. u × v = 12, –5, 7
D. u × v = 13, 5, 11
Example 3

21. Step 1 Graph each vector in standard position.
Torque Using Cross Product
MACHINERY A mechanic uses a 0.4-meter long wrench to tighten a nut. Find the magnitude and direction of the torque about the nut if the force is 30 newtons straight down to the end of the handle when it is 35° above the positive x-axis.
Step 1 Graph each vector in standard position.
Example 4

22. Step 2 Determine the component form of each vector.
Torque Using Cross Product
Step 2 Determine the component form of each vector.
The component form of the vector representing the directed distance from the axis of rotation to the end of the handle can be found using the triangle in the figure below and trigonometry.
Example 4

23. T = r × F Torque Cross Product Formula
Torque Using Cross Product
Vector r is therefore 0.4 cos 35°, 0, 0.4 sin 35° or about 0.33, 0, 0.23. The vector representing the force applied to the end of the handle is 30 newtons straight down, so F = 0, 0, –30.
Step 3 Use the cross product of these vectors to find the vector representing the torque about the nut.
T = r × F Torque Cross Product Formula
Cross product of r and F
Example 4

24. Determinants of 2 × 2 matrices
Torque Using Cross Product
Determinant of a 3 × 3 matrix
Determinants of 2 × 2 matrices
Component form
Example 4

25. Step 4 Find the magnitude and direction of the torque vector.
Torque Using Cross Product
Step 4 Find the magnitude and direction of the torque vector.
The component form of the torque vector 0, 9.9, 0 tells us that the magnitude of the vector is about 9.9 newton-meters parallel to the positive y-axis as shown below.
Answer: 9.9 N • m parallel to the positive y-axis
Example 4

26. A. 10.5 N • m parallel to the positive y-axis
MACHINERY A mechanic uses a 0.3-meter-long wrench to tighten a nut. Find the magnitude and direction of the torque about the nut if the force is 35 newtons straight down to the end of the handle when it is 40° below the positive x-axis as shown below.
A N • m parallel to the positive y-axis
B. 8.0 N • m parallel to the positive y-axis
C. 6.7 N • m parallel to the positive y-axis
D. 4.1 N • m parallel to the positive y-axis
Example 4

27. Determinant of a 3 × 3 matrix
Area of a Parallelogram in Space
Find the area of the parallelogram with adjacent sides u = –3i – 4j +2k and v = 5i – 4j – k.
Step 1 Find u × v.
u = –3i – 4j +2k and v = 5i – 4j – k
Determinant of a 3 × 3 matrix
Example 5

28. Determinants of 2 × 2 matrices
Area of a Parallelogram in Space
Determinants of 2 × 2 matrices
Step 2 Find the magnitude of u × v.
Magnitude of a vector in space
Simplify.
Example 5

29. The area of the parallelogram shown is or about 34.9 square units.
Area of a Parallelogram in Space
The area of the parallelogram shown is or about 34.9 square units.
Answer:
Example 5

30. Find the area of a parallelogram with sides u = 4i + 5j – 2k and v = i – j + 3k.
A. about 7.3 square units
B. about 10.0 square unit
C. about 20.8 square units
D. about 21.1 square units
Example 5

31. Key Concept 6

32. t = –3i + 3j + 2k , u = –3i – 4j + 2k and v = 5i – 4j – k
Volume of a Parallelepiped
Find the volume of the parallelepiped with adjacent edges t = –3i + 3j + 2k, u = –3i – 4j + 2k, and v = 5i – 4j – k.
t = –3i + 3j + 2k , u = –3i – 4j + 2k and v = 5i – 4j – k
Determinant of a 3 × 3 matrix
Example 6

33. Determinants of 2 × 2 matrices
Volume of a Parallelepiped
Determinants of 2 × 2 matrices
Simplify.
The volume of the parallelepiped shown below is | t ● (u × v)| or 49 cubic units.
Answer: 49 cubic units
Example 6

34. Find the volume of the parallelepiped with adjacent sides t = 2i – j + k, u = i + 2j – k, and v = 2i + 3j – k.
A. 2 cubic units
B. 6 cubic units
C. 8 cubic units
D. 14 cubic units
Example 6

35. Dots and Cross Products of Vectors in Space
LESSON 8–5
Dots and Cross Products of Vectors in Space