1/25 Recitation Quiz 1 Prep: Definitions, Theorems, and Proofs Flashcards
Show that vector u has length 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
What’s the trig identity for cos and sin that can be manipulated for various types of problems?
cos^2theta + sin^2theta = 1
Dot product and vectors orthogonal theorem
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
Law of cosines
c=sqrt(a^2+b^2﹣2abcostheta)
What is the equation for scalar projection of vector b onto vector a?
comp(small)a(large)b = vector a & b dot prod/magnitude of a
What is the equation for vector projection of vector b onto vector a?
proj(small)a(large)b = (vector a & b dot prod/(magnitude of a)^2)* vector a
Property of dot product 1: dot product of vector a* vector a
a * a = <a1, a2> * <a1, a2> = a1^2 + a2^2 = (magnitude of a)^2
Property of dot product 2: dot product of vector a * vector b
=dot product of vector b * vector a
Property of dot product 3: (ca(a is vector) dot prod. vector b
= c * (ab) = a * (cb(
proof:
look at picture
Property of dot product 4: vector 0 dot product vector a
= <0, 0> * <a1, a2> = 0 + 0 = 0
Property of dot product 5: vector a dot prod. (b + c (both vectors))
= vector a dot prod. vector b + vector a dot prod. vector c
An example of a parametric curve made of parametric equations
{x=x(t)
{y=y(t)
Vector function definition
A function of the form r(t) = <x(t), y(t)> is called a vector function
r(t) = x(t)i + y(t)j
A vector function describing a circle of radius a
r(t) = acost, asint>
Vector equation of a line
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
Parametric equations of a line
x(t) = x0 +at
y(t) = y0 + bt
Limit definition
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)
One-sided limits definition
limit from the left
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
One-sided limits definition
limit from the right
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
lim (x->a) f(x) = L <=> lim (x->a+) f(x) = lim (x->a-) f(x) = L
Limit lim (x->a) f(x) exists and equals L if and only if both one-sided limits exist and also equal L
Infinite limits definition
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)
Infinite limits for one-sided
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
Vector orthogonal complement definition
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
