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MATH 32A- Final Exam Guide - Comprehensive Notes for the exam ( 26 pages long!)


Department
Mathematics
Course Code
MATH 32A
Professor
Austin, A.D.
Study Guide
Final

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UCLA
MATH 32A
Final EXAM
STUDY GUIDE

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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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