1. Precalculus PreAP/Dual, Revised ©2017
Vectors
Section 6.3
Precalculus PreAP/Dual, Revised ©2017
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§6.3: Vectors in the Plane

2. Definitions
Vector is used to indicate a quantity that has both magnitude (length/distance) and direction.
Magnitude Equation: ∥𝑃𝑄∥= 𝑥 2 − 𝑥 𝑦 2 − 𝑦 1 2
Slope: 𝒎= 𝒚 𝟐 − 𝒚 𝟏 𝒙 𝟐 − 𝒙 𝟏
Component Form Equation is 𝒗= 𝒙 𝟐 − 𝒙 𝟏 , 𝒚 𝟐 − 𝒚 𝟏
Represented by an arrow or a directed line segment
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§6.3: Vectors in the Plane

3. Vector Example Vector: has magnitude (length) and direction
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§6.3: Vectors in the Plane

4. Example 1
Find the magnitude of vector 𝑷𝑸 where 𝑷 𝟐, 𝟑 and 𝑸 𝟓, 𝟗 and its component form.
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§6.3: Vectors in the Plane

5. Example 1
Find the magnitude of vector 𝑷𝑸 where 𝑷 𝟐, 𝟑 and 𝑸 𝟓, 𝟗 and its component form
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§6.3: Vectors in the Plane

6. Example 1
Find the magnitude of vector 𝑷𝑸 where 𝑷 𝟐, 𝟑 and 𝑸 𝟓, 𝟗 and its component form
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§6.3: Vectors in the Plane

7. Example 2
Show that 𝒖 and 𝒗 are equivalent through magnitude, slope and vector form of 𝒗
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§6.3: Vectors in the Plane

8. Your Turn
Find the magnitude of vector 𝑷𝑸 where 𝑷 −𝟑, 𝟓 and 𝑸 𝟕, −𝟏𝟏 and component form.
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§6.3: Vectors in the Plane

9. Vector Operations
Geometrically, the product of a vector 𝒗 and scalar 𝒌 is the vector that is 𝒌 times as long as 𝒗
If k is positive, 𝒌𝒗 has the same direction as 𝒗, and if 𝒌 is negative, 𝒌𝒗 has the opposite direction.
Equivalent Vectors is where every vector is equal to another vector with the initial point at the origin.
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§6.3: Vectors in the Plane

10. Example 3
If 𝒖= 𝟏,𝟔 and 𝒗= −𝟒,𝟐 , solve for 𝒖+𝒗 and 𝟑𝒗 and find the magnitude of 𝒖+𝒗
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§6.3: Vectors in the Plane

11. Example 3
If 𝒖= 𝟏,𝟔 and 𝒗= −𝟒,𝟐 , solve for 𝒖+𝒗 and 𝟑𝒗 and find the magnitude of 𝒖+𝒗
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§6.3: Vectors in the Plane

12. Your Turn
If 𝒖= 𝟑,𝟏 and 𝒗= −𝟖, 𝟒 , solve for 𝟐𝒖−𝟑𝒗 and find the magnitude of 𝟐𝒖−𝟑𝒗
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§6.3: Vectors in the Plane

13. Finding the Unit Vector
Have a magnitude of 1
Solve for the magnitude
Unit Vector equation: 𝒖= 𝒗 𝒗 = 𝒗 𝒙 𝟐 + 𝒚 𝟐
To find a unit vector 𝒖 that has the same direction as vector 𝒗: 𝒖= 𝒗 𝒗 = 𝒙 𝟏 𝒗 , 𝒚 𝟏 𝒗
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§6.3: Vectors in the Plane

14. Example 4 Find the unit vector 𝒗= −𝟐,𝟓 12/4/2018 4:14 PM
§6.3: Vectors in the Plane

15. Example 4 Find the unit vector 𝒗= −𝟐,𝟓 12/4/2018 4:14 PM
§6.3: Vectors in the Plane

16. Example 5 Find the unit vector of 𝒗= 𝟕,−𝟑 12/4/2018 4:14 PM
§6.3: Vectors in the Plane

17. Your Turn Find the unit vector of 𝒗= 𝟔,−𝟏 12/4/2018 4:14 PM
§6.3: Vectors in the Plane

18. Linear Combination/Alternate Notation
Vectors 𝒗= 𝒗 𝟏 , 𝒗 𝟐 is also represented as 𝒗=𝒊+𝒋
𝒊 and 𝒋 are considered component vectors whereas 𝒙𝒊+𝒚𝒋 is a linear combination of 𝒊 and 𝒋
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§6.3: Vectors in the Plane

19. Example 6
The initial point of a vector is 𝟎, −𝟐 and the terminal point is 𝟑, 𝟔 . Write a linear combination of the standard vector.
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§6.3: Vectors in the Plane

20. Your Turn
The initial point of a vector is −𝟐, 𝟔 and the terminal point is −𝟖, 𝟑 . Write a linear combination of the standard vector.
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§6.3: Vectors in the Plane

21. Direction Angles
Direction Angle: measured counterclockwise from the 𝒙–axis to terminal point of 𝒖
𝒖= 𝒙,𝒚 = 𝐜𝐨𝐬⁡𝜽, 𝐬𝐢𝐧⁡𝜽 = 𝐜𝐨𝐬 𝜽 𝒊+ 𝐬𝐢𝐧 𝜽 𝒋
𝒗=||𝒗||= 𝐜𝐨𝐬⁡𝜽, 𝐬𝐢𝐧⁡𝜽 = ||𝒗||𝐜𝐨𝐬 𝜽 𝒊+||𝒗|| 𝐬𝐢𝐧 𝜽 𝒋
Since 𝒗=𝒂𝒊+𝒃𝒋, 𝐭𝐚𝐧 𝜽 = 𝒃 𝒂 and use reference angles and Unit Circle when possible
When finding angles, remember the reference angle rules
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§6.3: Vectors in the Plane

22. Example 7 Find the direction angle and magnitude of 𝒖=𝟑𝒊+𝟑𝒋
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§6.3: Vectors in the Plane

23. Example 8 Find the direction angle of 𝒖=𝟑𝒊−𝟒𝒋 12/4/2018 4:14 PM
§6.3: Vectors in the Plane

24. Your Turn Find the direction angle of 𝒗=−𝟒𝒊 −𝟓𝒋 12/4/2018 4:14 PM
§6.3: Vectors in the Plane

25. Example 9
Find the component form of a vector that represents the velocity of an airplane descending at a speed of 150 miles per hour at an angle of 𝟐𝟎° below the horizontal.
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§6.3: Vectors in the Plane

26. Your Turn
Find the component form of the velocity vector that represents an airplane descending West at a speed of 100 mph at an angle of 𝟑𝟎° below horizontal.
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§6.3: Vectors in the Plane

27. More Resultant Force Steps
The wind pushes it in another direction. That is another vector.
The resultant vector is the ground speed.
Identify the two forces and then add them
𝒖=𝐚𝒊+𝐛𝒋 represents the vector where ||𝒗||𝐜𝐨𝐬 𝜽 𝒊+||𝒗|| 𝐬𝐢𝐧 𝜽 𝒋
Identify the magnitude and direction using the appropriate equations
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§6.3: Vectors in the Plane

28. Example 10
A Boeing 727 airplane, flying due east at 500 mph in still air, encounters a 70–mph tail wind acting in the direction of 60° north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N E v u
60o N E u
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§6.3: Vectors in the Plane

29. Example 10
A Boeing 727 airplane, flying due east at 500 mph in still air, encounters a 70–mph tail wind acting in the direction of 60° north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N
Need to find the magnitude and direction of the resultant vector u + v. v 70
u + v E
500 u
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§6.3: Vectors in the Plane

30. Example 10
A Boeing 727 airplane, flying due east at 500 mph in still air, encounters a 70–mph tail wind acting in the direction of 60° north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they?
12/4/2018 4:14 PM
§6.3: Vectors in the Plane

31. Example 10
A Boeing 727 airplane, flying due east at 500 mph in still air, encounters a 70–mph tail wind acting in the direction of 60° north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N E
538.4
6.5o N E v u
60o N E u
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§6.3: Vectors in the Plane

32. Assignment
Page odd, odd (DO NOT SKETCH), odd, odd, odd
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§6.3: Vectors in the Plane