Midterm 1

Question 1

Facts about cross multiplication of i j k

Answer 1

i x j = k vs j x i = -k
j x k = i vs k x j = -i
k x i = j vs i x k = -j
i x i = j x j = k x k = 0

Question 2

Vector projections and scalar projections (a onto b)

Answer 2

Vector: a dot b / b dot b x b
scalar: a dot b / |b|

Question 3

Area of a parallelagram and thus half that (a triangle)

Answer 3

Calculate AB and AC
mag(AB x AC) = Parallelogram
1/2 * that = Triangle

Question 4

Limits of functions with x and y where x, y -> (0,0)

Answer 4

Choose different paths (x=y, x=y^2) and if you’re able to find two different limits then the limit DNE, otherwise try to use identities or squeeze theorem

Question 5

angle formula

Answer 5

cos(theta) = (a dot b) / (|a||b|)

Question 6

scalar triple product

Answer 6

a⋅(b x c)
Determinant of
a1 a2 a3
b1 b2 b3
c1 c2 c3

Question 7

The distance between two points

Answer 7

The difference between each point
sqrt((a2 - a1)^2 + (b2 - b1)^2)

Question 8

Position vector

Answer 8

Starts from the origin

Question 9

Comp b (a)

Answer 9

|a|cos(theta) = (a dot b) / |b|

Question 10

Cross product of 3d vectors

Answer 10

⟨a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1⟩

Question 11

Another way of determining a x b

Answer 11

Det(
i j k
a1 a2 a3
b1 b2 b3
)

Question 12

Properties of a x b

Answer 12

The vector produced by a x b is orthogonal to both a and b
mag(a x b) = mag(a)⋅mag(b)⋅sin(theta)
if a x b = 0, then the two lines are parallel

Question 13

Ellipsiod general formula

Answer 13

d = x^2 + y^2 + z^2
like a FOOTBALL

Question 14

Elliptic paraboloid

Answer 14

z = x^2 + y^2
Like a cup

Question 15

Hyperbolic Paraboloid

Answer 15

z = x^2 - y^2
Like a saddle

Question 16

Cone

Answer 16

z^2 = x^2 + y^2
Dual bladed ice cream cone

Question 17

Hyperboloid of One Sheet

Answer 17

x^2 + y^2 - z^2 = 1
Like a bad yo yo on one of those free yo yos

Question 18

Hyperboloid of two sheets

Answer 18

-x^2 - y^2 + z^2 = 1
A popper right before and after launch

Question 19

Arc length of a curve

Answer 19

s(t) = integral a -> t |r’(u)|du

Question 20

Meaning of the following vectors on a curve:
Tangent
Normal
Binormal

Answer 20

What direction the curve is going
What direction the curve is turning towards
A line orthogonal to both of those vectors

Question 21

Acceleration in terms of tangential and normal acceleration

Answer 21

at T + an N

Question 22

What is a level curve?

Answer 22

a curve with the equation 𝑓(𝑥, 𝑦) = 𝑘

Question 23

What is the collection of level curves called?

Answer 23

A contour map (kind of like topology)

Question 24

formula for a sphere

Answer 24

(x-a)^2 + (y-b)^2 + (z-c)^2 = r^2 where the center of the sphere is at (a, b, c) and the radius is r

Question 25

Equation for a plane in terms of x, y, z

Answer 25

If (a, b, c) is a point on the plane and t(j, k, l) is the direcition part (the part that can be orthogonal) then j(x - a) + k(y - b) + l(z - c) = 0