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Vectors Math 200 Week 1- Wednesday Math 200 Main Questions for Today Vectors Math 200 Week 1- Wednesday Math 200 Main Questions for Today

Vectors Math 200 Week 1- Wednesday Math 200 Main Questions for Today - PowerPoint Presentation

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Vectors Math 200 Week 1- Wednesday Math 200 Main Questions for Today - PPT Presentation

Vectors Math 200 Week 1 Wednesday Math 200 Main Questions for Today What is a vector What operations can we do with vectors What kinds of representations and notations do we have for vectors What are some of the important properties of vectors ID: 762316

200 vectors vector math vectors 200 math vector magnitude scalar direction unit parallel length position

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Vectors Math 200 Week 1- Wednesday

Math 200 Main Questions for Today What is a vector? What operations can we do with vectors? What kinds of representations and notations do we have for vectors? What are some of the important properties of vectors?

Math 200 Definition A vector is a mathematical object with magnitude and direction . Representation: arrow whose length corresponds to magnitude Notation: Bold letters - e.g. v as opposed to vArrow over letter:

Math 200 Sums of vectors We add two vectors by putting them tail-to-end and then drawing the resultant vector from start to finish. E.g., say we want to add two vectors v and w, shown below.

Math 200 ¡¡Important!! VECTORS DO NOT HAVE FIXED POSITION! THESE ARE ALL THE SAME BECAUSE THEY HAVE THE SAME DIRECTION AND MAGNITUDE

Math 200 Scalar Multiples of vectors A scalar is a quantity with only magnitude (no direction) By multiplying a vector by a scalar , we canScale its length/magnitude Change its direction if the scalar is negative

Math 200 Differences of vectors With scalars, we can think of subtraction as addition of negative numbers. E.g., 5 - 2 = 5 + (-2) = 3 The same goes for vectors: v - w = v + ( -w)

Math 200 Vectors in rectangular Coordinates Suppose I draw a vector from one point to another. Call these points P 1 (x 1 , y 1, z1 ) and P2(x 2, y2, z 2) P 1 P 2 We’ll write this vector in the following way:

Math 200 Standard Position When we draw a vector with its tail at the origin, we’ll say it’s in standard position. For a vector in standard position, the terminal point gives the components directly… So if this is the point (2,5,2), then the vector is…

Math 200 A 2d Example Do the following: Draw and label x and y axesPick two vectors (e.g. v = <1,2> and w = <-3,1>)Draw v with its tail at the originDraw w with its tail at the terminal point of vDraw v + w

Math 200 <1,2> <3,1> <1+3, 2+1> = <4,3>

Math 200 properties Vector operations work pretty much the way we’d like!

Math 200 Parallel Vectors We saw that scalar multiplication always gives us a parallel vector to the original. E.g. If v = <1,2,-4>. Then 2 v = <2, 4, -8> and -3v = <-3, -6, 12> are both parallel to v Conversely, we can say that two vectors are parallel if they are scalar multiples of one another!E.g. v = <1,3,1> and w = <-4, -12, -4> are parallel because w = -4vNon-example: v = <1,3,1> and q = <2, 6, 4> are not parallel.How do we know? q1 = 2v1 but q3 = 4v 3

Math 200 Norm of a vector The norm of a vector is just its magnitude/length. We write || v|| for the “norm of v”Since vectors don’t have a fixed position, we can put any vector in standard position and use those components in the distance formula: E.g.,

Math 200 What can we say about || k v ||? We know that k scales the magnitude of vWe also know that if k is negative, we flip the direction of v Let’s see if that intuition corresponds to the algebra:

Math 200 Unit vectors When a vector has length one, we call it a unit vector Later, we’ll use unit vectors to define something called the directional derivative Often, we’ll need to take a non-unit vector, and shrink it to unit length This is called normalizing a vector

Math 200 Normalizing vectors Suppose we want a vector in the same direction as a vector v = <2,-1,1>, but with length one. We know that scalar multiplication by a positive number preserves direction.Let’s multiply v by 1/||v ||

Math 200 Three special unit vectors The three simplest unit vectors: <1,0,0>, <0,1,0>, <0,0,1> These get special names: i = <1,0,0> j = <0,1,0> k = <0,0,1>An alternative notation for vectors uses i, j, k:E.g., <2,3,6> = 2i + 3j + 6k And sometimes they get hats

Math 200 An Application <0,-200> <Acos(3π/4),Asin(3π/4)> <Bcos(π/6),Bsin(π/6)> *Think of each of these vectors the hypotenuse of a right triangle. We want to write the x and y components.

Math 200 If the weight isn’t moving , the three force vectors should add up to <0,0> If two vectors are equal , they must have the same components .

Math 200 Elimination Substitution