8-2: Geometric Vectors & 8-3: Vectors in Three- Dimensional Space Objectives Find ordered pairs that represent vectors. Add, subtract, multiply and find the magnitude of vectors algebraically. Add and subtract vectors in three- dimensional space. Find the magnitude of vectors in three- dimensional space. Definition Vector: quantity that has both magnitude and direction A vector is represented geometrically by a directed line segment. Vector The length represents the magnitude of the vector. The direction of the arrowhead indicates the direction of the vector. Q P P: initial point Q: terminal point Vector: Magnitude: Standard Position If the initial point is at the origin, the vector is in standard position. The direction is the directed angle between the positive x-axis and the vector. The zero vector has its initial point and terminal point at the origin. The magnitude is 0, and it can have any direction. Example Use a ruler and protractor to determine the magnitude and the direction of Direction: 50 Magnitude: 4.75 in Equal Vectors Two vectors are equal iff they have the same direction and the same magnitude. Resultant The sum of two or more vectors is called the resultant of the vectors. Vector Notation Vectors can be represented as Finding Magnitude Pythagorean Theorem P 1 (x 1,y 1 ) P 2 (x 2,y 2 ) y 2 -y 1 x 2 -x 1 Representation as an Ordered Pair Let P 1 (x 1,y 1 ) be the initial point of a vector and P 2 (x 2,y 2 ) be the terminal point. The ordered pair that representsis Its magnitude is given by Example Write the ordered pair that represents the vector from C (7, -3) to D (-2, -1). Find the magnitude. Vector Operations The following operations are defined for Example Let Example Ms. Gonzalez is pushing a stretcher with a force of 135N at 58 with the horizontal, while Mr. Howard is pulling the stretcher with a force of 214N at 43 with the horizontal. What is the magnitude of the force exerted on the stretcher? Example (continued) Unit Vector Vector that has magnitude of one unit Sum of Unit Vectors Example 8-3 Vectors in Three Dimensional Space Vectors can also be in three-dimensional space. x y z x-axis is coming out of the screen Example Locate the point at (-4, 3, 2). x y z 3D Formulas Formulas are the same as 2D except you must consider the z-coordinate. Representation as an Ordered Triple Let P 1 (x 1,y 1,z 1 ) be the initial point of a vector and P 2 (x 2,y 2,z 2 ) be the terminal point. The ordered triple that represents is Its magnitude is given by Example Write the ordered triple that represents the vector from A(-2,5,0) and B(3,1,8). Example Find the magnitude of if C(0,2,5) and D(4,0,3). Example Find the ordered triple that represents Unit Vectors in 3D Unit vectors in three-dimensional space: on the x-, y-, and z-axes respectively. Example Homework 8-3: p. 503 #15-36 multiples of 3 #37