1. 10.6: Applications of Vectors in the Plane
Page 667 1) 3i
–i – 2j
–i – 5j 3)
i – 4j
7i – 4j
18i – 12j 5)
𝟑 𝒊+ 𝟐 𝒋
− 𝟑 𝒊+ 𝟐 𝒋
−𝟐 𝟑 𝒊+𝟑 𝟐 𝒋 7)
(5/2)i + 2j 9)
7i + 7j
11)
9i – 18j
13)
𝟓 𝟑 𝟐 , 𝟓 𝟐
15)
−𝟏𝟎,𝟏𝟎 𝟑
17)
−𝟕.𝟓𝟏𝟕𝟓,𝟐.𝟕𝟑𝟔𝟐
19)
𝟏.𝟗𝟐𝟖𝟒,−𝟐.𝟐𝟗𝟖𝟏
21)
||v|| = 10, θ = 60°
23)
||v|| = 𝟒𝟏 , θ≈ °
25)
||v|| =𝟒 𝟓 , θ≈ °
27)
||v|| =𝟓 𝟏𝟑 , θ≈ °
29)
− 𝟕 𝟏𝟏𝟑 𝟏𝟏𝟑 𝒊+ 𝟖 𝟏𝟏𝟑 𝟏𝟏𝟑 𝒋
31)
− 𝟏𝟎 𝟏𝟎 𝒊− 𝟑 𝟏𝟎 𝟏𝟎 𝒋
43)
Ground speed: mph, 154.3°
12/1/2018 7:37 PM
10.6: Applications of Vectors in the Plane

2. Pre-Calculus AB PreAP/Dual, Revised ©2014
Dot Product
Section 10.6A
Pre-Calculus AB PreAP/Dual, Revised ©2014
12/1/2018 7:37 PM
10.5: Vectors

3. Definitions The dot product (also called inner product) of two vectors
The product of u <u1, u2> and v <v1, v2> is u • v = u1v1 + u2v2 as it yields to a scalar (known as a number)
The product of u <u1, u2>, v <v1, v2>, and w <w1, w2> is u • v • w yields to a vector
12/1/2018 7:37 PM
10.5: Vectors

4. Example 1
Find the dot product of u • v where u = <4, 5> and v = <2, 3>
12/1/2018 7:37 PM
10.5: Vectors

5. Example 2
Find the dot product of u • v where u = 8i – 2j and v = 4i – 3j
12/1/2018 7:37 PM
10.5: Vectors

6. Your Turn
Find the dot product of u • v where u = 2i – 3j and v = i + 5j
12/1/2018 7:37 PM
10.5: Vectors

7. Example 3
Find the dot product of u • (v + w) where u = <2, 5>, v = <–4, 3>, and w = <2, –1>
12/1/2018 7:37 PM
10.5: Vectors

8. Example 3
Find the dot product of u • (v + w) where u = <2, 5>, v = <–4, 3>, and w = <2, –1>
12/1/2018 7:37 PM
10.5: Vectors

9. Your Turn
Find the dot product of u • 2v where u = <–2, 5> and v = <1, 2>
12/1/2018 7:37 PM
10.5: Vectors

10. The Angle between Two Vectors
If θ is the angle between two nonzero vectors u and v:
12/1/2018 7:37 PM
10.5: Vectors

11. Example 4 Find the angle between u = <4, 3> and v = <3, 5>
12/1/2018 7:37 PM
10.5: Vectors

12. Example 5 Find the angle between u = 2i – 3j and v = i – 2j
12/1/2018 7:37 PM
10.5: Vectors

13. Your Turn Find the angle between u = i – 3j and v = 2j
12/1/2018 7:37 PM
10.5: Vectors

14. Definition of Vectors Parallel: If the slopes are the same (y/x)
Orthogonal: If the slopes are perpendicular (dot product = 0)
Neither: the slopes are neither parallel or perpendicular
12/1/2018 7:37 PM
10.5: Vectors

15. Steps
Find the dot product first to determine whether it is orthogonal (perpendicular)
If not, apply the equations, v = ku and solve for k.
If the k on both equations are equal, the slope is parallel
If the k on both equations are not equal, the slope is neither.
12/1/2018 7:37 PM
10.5: Vectors

16. Example 6
Are the vectors orthogonal, parallel or neither: u = <2, –3> and v = <6, 4>
12/1/2018 7:37 PM
10.5: Vectors

17. Example 7
Are the vectors orthogonal, parallel or neither: u = <6, 3> and v = <8, 4>
12/1/2018 7:37 PM
10.5: Vectors

18. Example 8
Are the vectors orthogonal, parallel or neither: u = <4, 7> and v = <5, 1>
12/1/2018 7:37 PM
10.5: Vectors

19. Your Turn
Are the vectors orthogonal, parallel or neither: u = <3, –7> and v = <–9, 21>
12/1/2018 7:37 PM
10.5: Vectors

20. Example 9
Solve for k so that vectors are orthogonal: u = 3i – 2j and v = 4i + kj
12/1/2018 7:37 PM
10.5: Vectors

21. Your Turn
Solve for k so that vectors are orthogonal: u = –4i + 5j and v = 2i + 2kj
12/1/2018 7:37 PM
10.5: Vectors

22. Assignment Page 679 3-11 EOO, 13-27 odd 12/1/2018 7:37 PM
10.5: Vectors