Boise State Math 275 (Ultman) Worksheet 1.1: Introduction to Vectors From the Toolbox (what you need from previous classes) Know how the Cartesian coordinates a point in the plane (R 2 ) determine its location. A vector can be thought of in two different ways: Geometric Form: Vectors can be drawn as arrows from an initial (starting) point, to a terminal (ending) point. Algebraic Form: Vector can be expressed numerically, by subtracting the coordinates of the terminal point from the coordinates of the initial point. Two vectors are equivalent if they have the same geometric and algebraic forms. In this worksheet, you will: Compute the algebraic form of a vector given its initial and terminal points. Compare vectors in algebraic and geometric forms. Determine whether vectors are equivalent. Compare position vectors and displacement vectors. Practice sketching 3-dimensional vectors. Definitions Scalar: A number (in this class, a real number). Usually denoted by numbers, or by lower-case roman letters or Greek letters. Examples of scalars: 5, 1/2, 0, 2, π, a, λ Point: A location in space. Usually denoted by upper-case letter (non-bolded). Coordinate form given using parentheses. Examples of points: P = (1, 6), Q = (a, b, c) Vector: A directed line segment (arrow) from one point to a second point. Usually denoted by bold letters or letters with arrows over them. Algebraic form given using angle brackets. Examples of vectors: v = 1, 6, w = a, b, c, P Q = 0.25, 0.5, 0 = 0, 0
Boise State Math 275 (Ultman) Worksheet 1.1: Introduction to Vectors 1 Model 1: Vectors in Algebraic Form, and Equivalent Vectors DIAGRAM 1: Critical Thinking Questions In this section, you will practice writing vectors in algebraic form, and examine what it means for two vectors to be equivalent. Notation: The vector P 1 P 2 begins at the initial point (or tail) P 1 = (x 1, y 1 ) and ends at the terminal point (or head) P 2 = (x 2, y 2 ). The algebraic form (also called the component form) of this vector is P 1 P 2 = x 2 x 1, y 2 y 1. (Q1) Find the algebraic form of the vector P Q. (Q2) Find the algebraic form of the vector OR. (Q3) Find the algebraic form of the vector v that has initial point ( 4, 0) and terminal point (3, 2), and sketch this vector on Diagram 1. (Q4) Which (if any) of the following vectors has algebraic form 6, 2? RP QR SP P S None of these. (Q5) Compare the algebraic forms of the vectors P Q and QP, and complete the statement: P Q =, QP =, Statement: To reverse the direction of a vector while keeping the same magnitude (length), multiplying all components of the algebraic form of the original vector by.
Boise State Math 275 (Ultman) Worksheet 1.1: Introduction to Vectors 2 (Q6) Use your result from (Q5) to answer the following: Suppose a vector v = a, b has initial point P 1 and terminal point P 2. What is the algebraic form of the vector w that starts at P 2 and ends at P 1? (Q7) Two vectors are equivalent if they have the same algebraic form. Suppose AB is the vector that has initial point A = ( 4, 2) and terminal point B = ( 2, 0). On Diagram 1, sketch the vector AB, and any equivalent vectors from the list below: SO OQ (Q8) Find the terminal point C of the vector OC that is equivalent to the vector AB from (Q7), and sketch OC on Diagram 1. (O is the origin: O = (0, 0).) (Q9) In (Q7)-(Q9), you should ve found that the vectors AB, the vector P Q, and the vector OC all have the same algebraic form, so they are all equivalent. P Q Compare the sketched you made of these three vectors on Diagram 1. Which of the following statements do you think is true? a) Equivalent vectors have the same magnitudes (lengths), but may have different directions. b) Equivalent vectors have the same direction, but may have different magnitudes (lengths). OR RS c) Equivalent vectors have the same direction and the same magnitude. d) Being equivalent doesn t tell you anything about the direction or magnitude of two vectors. Model 2: Two Applications of Vectors: Position & Displacement DIAGRAM 2:
Boise State Math 275 (Ultman) Worksheet 1.1: Introduction to Vectors 3 Critical Thinking Questions In this section, you will compare position and displacement vectors. A position vector starts at the origin, and ends at a point P. It gives the position of an object located at position P relative to the origin O (it points to the point P ). A displacement vector begins at a point P and ends at another point Q. It indicates the change in position (displacement) of an object that moves from location P to location Q. (Q10) Complete the table using the points from Diagram 2, then sketch and label these position vectors on Diagram 2. Point Position Vector P = (2, 1) r = 2, 1 Q = (4, 1) r = = r = 3, 2 S = r = (Q11) Complete the table using the points from Diagram 2, then sketch and label these displacement vectors on Diagram 2. Initial Point Terminal Point Displacement Vector P = (2, 1) Q = (4, 1) r = P Q = 2, 2 P = (2, 1) S = ( 4, 3) r = P S = O = (0, 0) = r = O = 3, 2 P = (2, 1) P = (2, 1) r = P P = (Q12) In (Q10), you found that the position vector of the point R is r = 3, 2. In (Q11), you found that the displacement vector from the origin O to the point R is r = OR = 3, 2. What can you say about these two vectors? (a) They have the same / different algebraic form(s). (b) They have the same / different initial and terminal points. (c) They have the same / different direction(s), and the same / different magnitude(s).
Boise State Math 275 (Ultman) Worksheet 1.1: Introduction to Vectors 4 ( Q13) The position vector r = 3, 2 and the displacement vector r = OR = 3, 2 have the same algebraic form, the same direction, and the same magnitude. Are they the same vector, or are the different? Explain. Model 3: Sketching Vectors in 3-Dimensions We are going to sketch the coordinate axes, and the vector v = 1, 3, 2 in standard position: beginning at the origin O = (0, 0, 0), ending at the point (1, 3, 2).
Boise State Math 275 (Ultman) Worksheet 1.1: Introduction to Vectors 5 Summary Vector are objects that possess both direction and (length). Vectors, points, and scalars are (choose one): All the same thing. All different things. Vectors are sometimes represented using bold text: v. ( True or False ) Vectors are sometimes represented using arrows: v. ( True or False ) In algebraic form, vectors are sometimes represented using angle brackets v 1, v 2, where v1 and v 2 are scalars. ( True or False ) One very useful property of vectors is that they can be represented both algebraically (in terms of numbers the algebraic form) and geometrically (in terms of magnitude and direction). For example, equivalent vectors not only have the algebraic form (an algebraic condition); they also have the same and (geometric conditions). To change the direction of a vector, multiply its components by. The position vector indicates the location of a point relative to the origin O. The vector r = P Q indicates the change in position of an object as it moves from a point P to a point Q.