Geometric Vectors Precalculus Chapter 8 Sections 1 2

Geometric Vectors

Pre-calculus Chapter 8 Sections 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 2 of 22

Pre-calculus Chapter 8 Sections 1 & 2 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 • If both the initial point and the terminal point are at the origin, the vector is the zero vector and is denoted by. is 45°. • The magnitude is 0 and it can be any direction. 3 of 22

Pre-calculus Chapter 8 Sections 1 & 2 Equal Vectors • Two vectors are equal if and only if they have the same direction and the same magnitude. • are equal since they have same direction and • • • are equal. have the same direction but they have different directions, so 4 of 22

Pre-calculus Chapter 8 Sections 1 & 2 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: 5 of 22

Pre-calculus Chapter 8 Sections 1 & 2 • Example 1 a. the parallelogram method Find the sum b. the triangle method 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. 6 of 22

Pre-calculus Chapter 8 Sections 1 & 2 • Example 1 (cont) ü the parallelogram method Find the sum the triangle method 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. 7 of 22

Pre-calculus Chapter 8 Sections 1 & 2 Opposite Vectors Two vectors are opposites if they have the same magnitude and opposite directions. are opposites, as are The opposite of You can use opposite vectors to subtract vectors. 8 of 22

Pre-calculus Chapter 8 Sections 1 & 2 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 9 of 22

Pre-calculus Chapter 8 Sections 1 & 2 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 10 of 22

Pre-calculus Chapter 8 Sections 1 & 2 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. 11 of 22

Pre-calculus Chapter 8 Sections 1 & 2 Example 3 A cruise ship leaves port and sails for 8 o miles in a direction of 50° north of due east. Draw a picture and find the magnitude of the vertical and horizontal components. 12 of 22

Algebraic Vectors

Pre-calculus Chapter 8 Sections 1 & 2 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 14 of 22

Pre-calculus Chapter 8 Sections 1 & 2 Ordered Pairs …Vectors Assume that P 1 and P 2 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. 15 of 22

Pre-calculus Chapter 8 Sections 1 & 2 Example 1 Write the ordered pair that represents the vector from C(7, – 3) to D(– 2, – 1). Then find the magnitude of 16 of 22

Pre-calculus Chapter 8 Sections 1 & 2 Vector Operations When vectors are represented by ordered pair they can be easily added, subtracted or multiplied by a scalar. 17 of 22

Pre-calculus Chapter 8 Sections 1 & 2 Example 2 Let Find each of the following. 18 of 22

Pre-calculus Chapter 8 Sections 1 & 2 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? 120 n. 55° 200 n. y 1 Mr. R. Ms. J. x 1 40° y 2 SOH CAH TOA x 2 19 of 22

Pre-calculus Chapter 8 Sections 1 & 2 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 expressed as can be 20 of 22

Pre-calculus Chapter 8 Sections 1 & 2 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. 21 of 22