Department

MathematicsCourse Code

MATH 32AProfessor

Austin, A.D.Study Guide

FinalThis

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MATH 32A

Final EXAM

STUDY GUIDE

Only pages 1-3 are available for preview. Some parts have been intentionally blurred.

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Lecture 1: Vectors in 2 Dimensions

- This is chapter 13 in the textbook “Multivariable Calculus”.

- A vector is a directed line segment.

- This is usually represented as an arrow in 2-D or 3-D space from one point to

another.

- Can draw an arrow on a graph from coordinates (2,1) to (4,4), that is a vector.

This vector is denoted as v, with an arrow pointing to the right on top of the v.

This is the notation for a vector in this course.

- A vector can be denoted as any letter, but usually v. But it must have an arrow

pointing to the right on top of it to indicate it’s a vector.

- The vector from point P to point Q is written as PQ, and putting one arrow

pointing to the right above both letters.

- We only care about the direction and length of the vector.

- Usually we don’t think of vectors as having a fixed position or going specifically

going from one place to another.

- If two vectors have the same length and direction, we can consider them to be

the same.

- Say we have a vector going from point P(2,1) to Q(4,4) and then also have a

vector going from R(5,-1) to S(7,2).

- The vectors PQ and RS are the same. If you calculate the distance from P to Q

and from R to S using the Pythagorean theorem, they are the same distance,

and they also have the same direction (Going diagonally up-right, moving to the

right 2 units and up 3 units).

- So, PQ = RS. Therefore, vectors don’t have to be in the same position to be the

same.

- One way of writing a vector numerically is to write it as a certain displacement in

the x-direction and a certain displacement in the y-direction.

- These numbers are called the components of the vector, specifically the x and y

components of the vector.

- From P to Q in the example above, the x-component is 2 and the y-component is

3.

- IF the vector’s direction was reversed, the x-component would be -2 and the

y-component is -3.

- You can write vector PQ numerically as PQ (with arrow on top) = <2,3>.

Numerical notation is <x-component, y-component>.

- You can get x-component by doing x2 - x1 and y-component by y2 - y1. In vector

PQ, x-component = 4 - 2 = 2, y-component = 4 - 1 = 3.

- Example: Vector v goes from (2,3) to (4,1). X-component = x2 - x1 = 4 - 2 = 2,

Y-component = y2 - y1 = 1 - 3 = -2. V = <2,-2>.

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- Vector w goes from (-1, 1) to (-2, -4). X-component = -2 - (-1) = -1. Y-component

= -4 - 1 = -5. W = <-1,-5>.

- If components of vectors are equal, the vectors are the same.

- The length of a vector, also called its magnitude, can be calculated easily from its

components, use pythagorean theorem, length of vector = square root of

(x-component)2 + (y-component)2.

- Why? Because the x-component of the vector drawn as a horizontal line and the

y-component of the vector drawn as a vertical line forms a right triangle, with the

vector itself serving as the hypotenuse.

- Example: Vector v goes from (0,0) to (5,-2). The length of the vector is square

root of 52 + (-2)2 = 29. The length of the vector is the square root of 29.

- ||v|| = (length of v). This is the notation of the magnitude of v. There is also an

arrow on top of the “v” in the double bars.

- In the above example, the answer would be written as ||v|| = square root of 29.

- SO: If v = <a,b>, then ||v|| = square root of a2 + b2.

- Addition of vectors: Given two vectors (v and w): Remember that vectors aren’t in

fixed positions. Therefore, they can be moved around to a convenient place as

long as their length and direction remain the same.

- So, you can move vector w in the scenario above so that vector w starts where

vector v ends (tip to tail). The resulting vector that forms from adding the two

vectors then is the resultant vector that goes from the starting point (tail) of vector

v to the ending point of vector w (tip).

- The resultant vector is denoted as v + w, with arrows over both letters.

- So, you can add x-components and y-components of vectors to get the

components of the resulting vector.

- If v goes from (0,0) to (4,2) and w goes from (0,0) to (2,-2), then v + w = <4 + 2,

2 + -2> = <6,0>. So, if you move vector w so that it starts from where v ends (at

point (4,2)), then vector w would end at (6,0) and the resulting vector would just

be a horizontal line from (0,0) to (6,0).

- If v = <a,b> and w = <c,d>, then v + w = <a+c, b+d>.

- NOTE: For vector addition, v + w = w + v. <a+c = c+a, b+d = d+b>.

- You can also multiply a vector by a scalar (Scalar multiplication or scaling).

- Scalar has only magnitude, no direction. It is pretty much a number.

- If vector v is multiplied by 2, then the resulting vector would be vector v plus the

same vector v that starts where the vector v ends. In other words, it would be a

vector with double the length of v but going in the same direction as v. Direction

remains same, length multiplied by whatever the scalar number is.

- So, if v = <a,b>, then v x scalar = <a x scalar, b x scalar>.

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