1/25 Recitation Quiz 1 Prep: Definitions, Theorems, and Proofs

Question 1

Show that vector u has length 1

Answer 1

Check picture
You basically use the unit vector formula and then expand it until it has an x and y value, then take the magnitude of that and you should get sqrt(1), which is 1

Question 2

What’s the trig identity for cos and sin that can be manipulated for various types of problems?

Answer 2

cos^2theta + sin^2theta = 1

Question 3

Dot product and vectors orthogonal theorem

Answer 3

Vectors a and b are orthogonal <=> (means if and only if) dot product of vectors a and b is 0

Proof: Assume vectors a and b are orthogonal, then the dot product = |a||b|cos90° = 0 because cos90° is 0, making the answer 0

Question 4

Law of cosines

Answer 4

c=sqrt(a^2+b^2﹣2abcostheta)

Question 5

What is the equation for scalar projection of vector b onto vector a?

Answer 5

comp(small)a(large)b = vector a & b dot prod/magnitude of a

Question 6

What is the equation for vector projection of vector b onto vector a?

Answer 6

proj(small)a(large)b = (vector a & b dot prod/(magnitude of a)^2)* vector a

Question 7

Property of dot product 1: dot product of vector a* vector a

Answer 7

a * a = <a1, a2> * <a1, a2> = a1^2 + a2^2 = (magnitude of a)^2

Question 8

Property of dot product 2: dot product of vector a * vector b

Answer 8

=dot product of vector b * vector a

Question 9

Property of dot product 3: (ca(a is vector) dot prod. vector b

Answer 9

= c * (ab) = a * (cb(

proof:
look at picture

Question 10

Property of dot product 4: vector 0 dot product vector a

Answer 10

= <0, 0> * <a1, a2> = 0 + 0 = 0

Question 11

Property of dot product 5: vector a dot prod. (b + c (both vectors))

Answer 11

= vector a dot prod. vector b + vector a dot prod. vector c

Question 12

An example of a parametric curve made of parametric equations

Answer 12

{x=x(t)
{y=y(t)

Question 13

Vector function definition

Answer 13

A function of the form r(t) = <x(t), y(t)> is called a vector function
r(t) = x(t)i + y(t)j

Question 14

A vector function describing a circle of radius a

Answer 14

r(t) = acost, asint>

Question 15

Vector equation of a line

Answer 15

A line L is given by the vector function r(t) = r0 + tv where v = any vector parallel to L
r0 = some fixed vector from (0,0) to L

Question 16

Parametric equations of a line

Answer 16

x(t) = x0 +at
y(t) = y0 + bt

Question 17

Limit definition

Answer 17

Suppose f(x) is defined on an open interval containing x=a (except possibly a itself) Then we say that lim (x->a) f(x) = L if we can make the values of f(x) arbitrarily close to L by taking x sufficiently close to a (but not equal to a)

Question 18

One-sided limits definition
limit from the left

Answer 18

lim (x->a-) f(x) = L
If we can make f(x) arbitrarily close to L by taking x sufficiently close to a and x<a

Question 19

One-sided limits definition
limit from the right

Answer 19

lim (x->a+) f(x) = L
If we can make f(x) arbitrarily close to L by taking x sufficiently close to a and x>a

Question 20

lim (x->a) f(x) = L <=> lim (x->a+) f(x) = lim (x->a-) f(x) = L

Answer 20

Limit lim (x->a) f(x) exists and equals L if and only if both one-sided limits exist and also equal L

Question 21

Infinite limits definition

Answer 21

lim(x->a) f(x) = ∞ means that the values of f(x) can be made arbitrarily large by taking x sufficiently close to a (but not equal to a)

lim(x->a) f(x) = -∞ means that the values of f(x) can be made arbitrarily negative large by taking x sufficiently close to a (but not equal to a)

Question 22

Infinite limits for one-sided

Answer 22

lim(x->a+) f(x) = ∞ means that the values of f(x) can be made arbitrarily large by taking x sufficiently close to a and x > a

lim(x->a-) f(x) = ∞ means that the values of f(x) can be made arbitrarily large by taking x sufficiently close to a and x < a

Also -∞ means arbitrarily negative large

Question 23

Vector orthogonal complement definition

Answer 23

For a vector a = <a1, a2>, its orthogonal complement is a(some upside down t symbol) = <-a2, a1> It has the property: a dot prod a(with the weird upside down t) = <a1, a2> dot prod <-a2, a1> = -a1a2 + a2a1 = 0