2. Vectors, Operations, and the Dot Product

3. Basic Terminology A vector in the plane is a directed line segment.
Consider vector AB
A is called the initial point
B is called the terminal point
Magnitude: length of a vector, expressed as
The sum of two vectors is also a vector.
The vector sum A + B is called the resultant.

4. Vectors

5. Basic Terminology continued
A vector with its initial point at the origin is called a position vector.
A position vector u with its endpoint at the point (a, b) is written
The numbers a and b are the horizontal component and vertical component of vector u.
The positive angle between the x-axis and a position vector is the direction angle for the vector.

6. Vector <a,b>

7. Magnitude and Direction Angle of Vector
The magnitude (length) of vector u = is given by
The direction angle  satisfies where a  0.

8. Example: Magnitude & Direction
Find the magnitude and direction angle for
Magnitude:
Direction Angle:
Vector u has a positive horizontal component.
Vector u has a negative vertical component, placing the vector in quadrant ?.

9. Example: Magnitude & Direction
Find the magnitude and direction angle for
Magnitude:
Direction Angle:
Vector u has a positive horizontal component.
Vector u has a negative vertical component, placing the vector in quadrant IV.

10. Horizontal and Vertical Components
The horizontal and vertical components, respectively, of a vector u having magnitude |u| and direction angle  are given by
That is,

11. Example: Components
Vector w has magnitude 35.0 and direction angle 51.2. Find the horizontal and vertical components.

12. Example: Components
Vector w has magnitude 35.0 and direction angle 51.2. Find the horizontal and vertical components.
Therefore, w =
The horizontal component is 21.9, and the vertical component is 27.3.

13. Example
Write each vector in Figure 29 in the form

14. Example
Write each vector in Figure 29 in the form

15. Solutions

16. Vector Operations
For any real numbers a, b, c, d, and k,

17. Example: Vector Operations
a) 4v
b) 2u + v
c) 2u  3v

18. i, j Forms for Vectors A unit vector is a vector that has magnitude 1.
Dot Product
The dot product of two vectors
is denoted u • v, read “u dot v,” and given by

19. Example: Dot Product
Find each dot product.

20. Properties of the Dot Product
For all vectors u, v, and w and real numbers k, a) b) c) d) e) f)

21. Geometric Interpretation of Dot Product
If  is the angle between the two nonzero vectors u and v, where 0   180, then

22. Example: Angle Between
Find the angle  between the two vectors
By the geometric

23. u + v

24. 7.5
Applications of Vectors

25. Equilabrant

26. Example
Forces of 10 newtons and 50 newtons act on an object at right angles to each other. Find the magnitude of the resultant and the angle of the resultant makes with the larger force. 10 50  v

27. Example
Forces of 10 newtons and 50 newtons act on an object at right angles to each other. Find the magnitude of the resultant and the angle of the resultant makes with the larger force.
The resultant vector, v, has magnitude 51 and make an angle of 11.3 with the larger force. 10 50  v

28. Example
A vector w has a magnitude of 45 and rests on an incline of 20. Resolve the vector into its horizontal and vertical components. v u 45
20

29. Example
A vector w has a magnitude of 45 and rests on an incline of 20. Resolve the vector into its horizontal and vertical components.
The horizontal component is 42.3 and the vertical component is 15.4. v u 45
20

30. Example
A ship leaves port on a bearing of 28.0 and travels 8.20 mi. The ship then turns due east and travels 4.30 mi. How far is the ship from port? What is its bearing from port?

31. Example continued Vectors PA and AE represent the ship’s path.
Magnitude and bearing:

32. Example continued The ship is about 10.9 mi from port.
To find the bearing of the ship from port, find angle APE.
Add 20.4 to 28.0 to find that the bearing is 48.4.

33. Vector Problem