# Lesson 2: Vectors and the Dot Product

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Section 9.2–3Vectors and the Dot ProductMath 21aFebruary 6, 2008AnnouncementsThe MQC is open: Sun-Thu 8:30pm–10:30pm, SC 222Homework for Friday 2/8:Section 9.2: 4, 6, 26, 33, 34Section 9.3: 10, 18, 24, 25, 34Section 9.4: 1*OutlineVectorsAlgebra of VectorsComponentsStandard basis vectorsLengthThe Dot ProductWorkConceptPropertiesUsesWhat is a vector?DeﬁnitionA vector is something that has magnitude and directionWe denote vectors by boldface (v) or little arrows (v ). One isgood for print, one for scriptGiven two points A and B in ﬂatland or spaceland, the vectorwhich starts at A and ends at B is called the displacement−→vector AB.Two vectors are equal if they have the same magnitude anddirection (they need not overlap)BDvuACscalarvectorvectorscalarVector or scalar?DeﬁnitionA scalar is another name for a real number.ExampleWhich of these are vectors or scalars?(i) Cost of a theater ticket(ii) The current in a river(iii) The initial ﬂight path from Boston to New York(iv) The population of the worldVector or scalar?DeﬁnitionA scalar is another name for a real number.ExampleWhich of these are vectors or scalars?(i) Cost of a theater ticket scalar(ii) The current in a river vector(iii) The initial ﬂight path from Boston to New York vector(iv) The population of the world scalarVector additionDeﬁnitionIf u and v are vectors positioned so the initial point of v is theterminal point of u, the sum u + v is the vector whose initial pointis the initial point of u and whose terminal point is the terminalpoint of v.uvvu + vu + vvuuThe triangle lawThe parallelogram lawOpposite and diﬀerenceDeﬁnitionGiven vectors u and v,the opposite of v is the vector −v that has the same lengthas v but points in the opposite directionthe diﬀerence u − v is the sum u + (−v)vu−vu − v2v− 1 v2Scaling vectorsDeﬁnitionIf c is a nonzero scalar and v is a vector, the scalar multiple cv isthe vector whoselength is |c| times the length of vdirection is the same as v if c > 0 and opposite v if c < 0If c = 0, cv = 0.v− 1 v2Scaling vectorsDeﬁnitionIf c is a nonzero scalar and v is a vector, the scalar multiple cv isthe vector whoselength is |c| times the length of vdirection is the same as v if c > 0 and opposite v if c < 0If c = 0, cv = 0.2vv2vScaling vectorsDeﬁnitionIf c is a nonzero scalar and v is a vector, the scalar multiple cv isthe vector whoselength is |c| times the length of vdirection is the same as v if c > 0 and opposite v if c < 0If c = 0, cv = 0.v− 1 v2Document Outline
• Announcements
• Vectors
• Algebra of Vectors
• Components
• Standard basis vectors
• Length
• The Dot Product
• Work
• Concept
• Properties
• Uses