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Fun with Vectors. Definition. A vector is a quantity that has both magnitude and direction Examples?. v, v , or AB. Represented by an arrow. B (terminal point). A (initial point). If two vectors, u and v , have the same length and direction, we say they are equivalent. v. u. - PowerPoint PPT Presentation

Fun with Vectors

DefinitionA vector is a quantity that has both magnitude and directionExamples?

Represented by an arrowA (initial point)B (terminal point)

If two vectors, u and v, have the same length and direction, we say they are equivalent uv

abVector addition

abVector addition: a + b

abVector addition: a + ba+b

abVector addition: a + ba+b

Scalar Multiplicationa

Scalar Multiplicationa2a-a- a

Subtractionba

Subtractionba-b

Subtractionba-ba+(-b)

Subtractionba-ba+(-b)

Subtractionba-ba+(-b)If a and b share the same initial point, the vector a-b is the vectorfrom the terminal point of b to the terminal point of a

Lets put these on a coordinate systemWe can describe a vector by putting its initial point at the origin.

We denote this as a=

where (a1,a2) represent the terminal point

Graphically a=(a1,a2)

Given two points A=(x1,y1) and B=(x2,y2),

The vector v = AB is given by

v =

or in 3-space, v =

GraphicallyA=(-1,2)B=(2,3)v = =

Recall, a vector has direction and lengthDefinition: The magnitude of a vector v = is given by

Properties of VectorsSuppose a, band c are vectors, c and d are scalarsa+b=b+aa+(b+c)=(a+b)+ca+0=aa+(-a)=0c(a+b)=ca+cb(c+d)a=ca+da(cd)a=c(da)1a=a

Standard Basis VectorsDefinition: vectors with length 1 are called unit vectors

Example: We can express vectors in terms of this basisa = a = 2i -4j+6kQ. How do we find a unit vector in the same direction as a?A. Scale a by its magnitude

Examplea =

12.3 The Dot ProductMotivation: Work = Force* Distance BoxFDFxFy

To find the work done in moving the box, we want the part of F in the direction of the distance

One interpretation of the dot productWhere is the angle between F and D

A more useful definitionYou can show these two definitions are equal by considering the following triangle and applying the law of cosines! See page 808 for detailsba-baThink, what is |a|2?

Examplea=, b=Find a.b and the angle between a and b

The Dot ProductIf a = and b= then

The dot product of a and b is a NUMB3R given by

The Dot Producta and b are orthogonal if and only if the dot product of a and b is 0Other Remarks:ab

Properties of the dot productSuppose a, b, and c are vectors and c is a scalara.a=|a|2a.b=b.aa.(b+c) = (a.b)+(a.c)(ca).b=c(a.b)=a.(cb)0.a=0

Yet another use of the dot product: ProjectionsThink of our work example: this is how much of b is in the direction of a

We call this quantity the scalar projection of b on aThink of it this way: The scalar projection is the length of the shadow of b cast upon a by a light directly above a

Q. How do we get the vector in the direction of a with length compab?We need to multiply the unit vector in the direction of a by compab.

We call this the vector projection of b onto a

Examples/Practice!

Key PointsVector algebra: addition, subtraction, scalar multiplicationGeometric interpretationUnit vectorsThe dot product and the angle between vectorsProjections (algebraic and geometric)