1. Vectors

2. Vectors Vectors describe magnitude and direction of a quantity.
A vector can be represented by a directed line segment with an initial point and a terminal point.
Vectors can be written as

3. Vectors
The magnitude of , denoted , is the length of the vector from its initial point to its terminal point.
The direction of a vector can be expressed as the angle it forms with the horizontal or as a measurement between 0 and 90 east or west of the north-south line.

4. Resultant
A resultant is the sum of two or more vectors.

5. Example
Copy the vectors. Then find

6. Example
Copy the vectors. Then find

7. Vectors on the Coordinate Plane
A vector is in standard position if its initial point is at the origin and can be described by its terminal point P(x, y).
To describe a vector with any initial point, you can use the component form ⟨x, y⟩, which describes the vector in terms of its horizontal component x and vertical component y.
To write the component form of a vector with initial point (x1, y1) and terminal point (x2, y2), find ⟨x2 – x1, y2 – y1⟩

8. Examples
Write the component form of vector CD.

9. Examples
Write the component form of vector PQ.

10. Magnitude and Direction
The magnitude of a vector can be found by using the Distance Formula, and the direction can be found by using trigonometric ratios.

11. Examples
Find the magnitude and direction of = ⟨-1,4⟩

12. Examples Find the magnitude and direction of = ⟨-1,4⟩ Magnitude:
or about 4.12

13. Examples Find the magnitude and direction of = ⟨-1,4⟩ Direction
Graph and its components.
The angle is the inverse tangent of the components.
θ = tan-1(1/4); θ = 14˚
The direction of is the angle that it makes with the positive x-axis, which is about 104˚.

14. Examples

15. Vector Operations

16. Examples
Find for , , and .

17. Examples
Find for , , and .
⟨1 - 3, ⟩
⟨-2, -6⟩