1. Geometric Vectors Chapter 8 SEC 1

2. Vector • A vector is a quantity that has both direction and magnitude. • It is represented geometrical by a directed line segment. • So a directed line segment with initial point P and terminal point Q denoted by • Direction of the arrowhead show direction. • The length represents the magnitude denoted by . Q P

3. Vector • If a vector has it’s initial point at the origin, it is in standard position. • The direction of the vector is the directed angle between the positive x-axis and the vector. • The direction of is 45°. • If both the initial point and the terminal point are at the origin, the vector is the zero vector and is denoted by . • The magnitude is 0 and it can be any direction.

4. Equal Vectors • Two vectors are equal if and only if they have the same directionand the same magnitude. • are equal since theyhave same direction and • are equal. • have the same direction but • but they have different directions, so

5. Resultant Vectors (sum of vectors) • The sum of two or more vectors is call the resultant of the vectors. The follow will give the resultant:

6. Example 1 the parallelogram method the triangle method • Find the sum a. Copy then copy placing the initial points together. Form a parallelogram and draw dashed lines for the other two sides. The resultant is the vector from the vertex to the opposite vertex.

7. Example 1 (cont) • the parallelogram method • the triangle method • Find the sum b. Copy then copy so the initial point of is on the terminal side of . The resultant is the vector from the initial point of to the terminal point of .

8. Opposite Vectors • Two vectors are oppositesif they have the same magnitude and opposite directions. • are opposites, as are • The opposite of • You can use opposite vectorsto subtract vectors.

9. Scalars • A quantity with only magnitude is called a scalar quantity. • Mass, time, length and temperature • The numbers used to measure scalar quantities are called scalars. • The product of a scalar k and a vector is a vector with the same direction as and a magnitude of , if k > 0. • If k < 0, the vector has the opposite direction of and a magnitude of

10. Example 2 Use the triangle method to find Rewrite Draw a vector twice the magnitude of Draw a vector opposite direction and half the magnitude of Put them tip to tail. Connect the initial point from and terminal point of

11. Parallel • Two or more vectors are said to be parallel if and only if they have the same or opposite direction. • Two or more vectors whose sum is a given vector are called components of the given resultant vector. • Often it is useful to have components that are perpendicular.

12. Example 3 • A cruise ship leaves port and sails for 8o miles in a direction of 50° north of due east. Draw a picture and find the magnitude of the vertical and horizontal components.

13. Algebraic Vectors Chapter 8 Sec 2

14. Algebraic Vectors • Vector can be represented algebraically using ordered pairs of real numbers. • The ordered pair (3, 5) can represent a vector in standard position. Initial pt (0,0) and terminal pt (3,5). • Horizontal Mag. of 3, Vertical Mag. of 5. • Since vector of same mag. and direction are equal many vectors can be represented by the same ordered pair. • In other words, a vector does not have to be in standard position to use ordered pairs

15. Ordered Pairs …Vectors • Assume that P1 and P2 are any two points. • Drawing the horizontal and vertical components yields a right triangle. • So, the magnitude can be found by using the Pythagorean Theorem.

16. Example 1 Write the ordered pair that represents the vector from C(7, – 3) to D(–2, –1). Then find the magnitude of

17. Vector Operations • When vectors are represented by ordered pair they can be easily added, subtracted or multiplied by a scalar.

18. Example 2 Let Find each of the following.

19. Example 3 Radiology technicians Mary Jones and Joseph Rodriguez are moving a patient on an MRI machine cot. Ms. J is pushing the cot with a force of 120 newtons at 55° with the horizontal, while Mr. R. is pulling the cot with a force of 200 newtons at 40° with the horizontal. What is the magnitude of the force exerted on the cot? SOH CAH TOA 200 n. 120 n. y1 y2 Mr. R. Ms. J. 55° x2 40° x1

20. Unit Vector • A vectors with a magnitude of one unit is called a unit vector. • A unit vector in the direction of the x–axis is represented by • A unit vector in the direction of the y–axis is represented by • So, • Any vectors can be expressed as

21. Example 4 Write as the sum of unit vectors for A(2, –7) and B(–1, 5). First write as an ordered pair. The write as the sum of unit vectors.

22. Daily Assignment • Chapter 8 Sections 1 & 2 • Text Book • Pg 491 • #15 – 25 Odd; • Pg 497 • #13 – 41 Odd;