Final

Question 1

Unit Vector

Answer 1

V / |V|

Question 2

Vector of a certain length in the direction of another vector

Answer 2

(V / |V| ) * length

Question 3

Angle Between Two Vectors

Answer 3

cos(x) = (a.b) / |a||b|

Question 4

Projection of Vector U onto V (vector)

Answer 4

[V.U / |V|^2 ] * V

Question 5

Projection of Vector U onto V (scalar)

Answer 5

V.U / |V|^2

Question 6

Area of a Parallelogram

Answer 6

|U * V|

Question 7

Volume of a Parallelpipid

Answer 7

(a * b) . c

Question 8

Requirements for Vectors Being Parallel

Answer 8

1) a = <b>c OR
2) a * b = 0 OR
3) a.b = |a||b|</b>

Question 9

Vectors are Coplanar

Answer 9

Triple Scalar Product = 0

Question 10

Parametric

Answer 10

<x,y,z> = <xo, yo, zo> + t<a,b,c>
x = xo + at
y = yo + bt
z = zo + ct

Question 11

Symmetric

Answer 11

t = (x-xo)/a = (y-yo)/b = (z-zo)/c

If a=0, then x=xo

Question 12

Length of a Curve

Answer 12

L = (int)_a^b |r’(t)|dt OR
L = (int)_a^B sqrt(f’(t)^2+g’(t)^2+h’(t)^2)dt

Question 13

Re-parametrize the curve using the length formula

Answer 13

• Set the found length equal to s
• Solve for t
• Plug t into the original r(t) equation

Question 14

Normal Vector

Answer 14

Of Two Parallel Planes: The slopes
Of Two Vectors: Cross product

Question 15

Binormal Vector

Answer 15

T(t) = r’(t) / |r’(t)|

Question 16

Projectile Motion

Answer 16

a(t) = -gj

Question 17

Equation of a Tangent Plane

Answer 17

z = zo + fx(xo,yo)(x-xo) + fy(xo,yo)(y-yo)

Question 18

Area of Triangle

Answer 18

A = 1/2 |B * H| (CROSS PRODUCT)

Question 19

Orthogonal when

Answer 19

dot product = 0

Question 20

Gradient

Answer 20

F = <fx, fy, fz>

Question 21

Gradient at point p

Answer 21

find gradient then plug in values

Question 22

Directional Derivative

Answer 22

Gradient * u

Question 23

Directional Derivative with given angle

Answer 23

Duf(x, y) = fx(x, y) cos(𝜃) + fy(x, y) sin(𝜃),

Question 24

Implicit Differentiation

Answer 24

(Df/Dx) * (Dx/Du) + (Df/Dy) * (Dy/Du) (change all values to the given value at the end)

Question 25

Max Value

Answer 25

D(a,b) > 0 and fxx(a,b) < 0
(a,b) is a maximum

Question 26

Min Value

Answer 26

D(a,b) > 0 and fxx(a,b) > 0
(a,b) is a mininum

Question 27

How to find min and max values

Question 28

Critical Point is when

Answer 28

the gradient of f is equal to 0 OR the partial derivative does not exist

Question 29

D(a,b) = 0 means

Answer 29

there is not enough information

Question 30

How to find critical points

Answer 30

Find the partial derivative of each (x,y,z), then set the equation equal to zero.

Question 31

How to find Min and Max values

Answer 31

D(a, b) = fxx(a, b)fyy(a, b) − (fxy(a, b))2

Question 32

D(a,b) is a saddle point when

Answer 32

D(a, b) < 0

Question 33

The procedure to find the maximum and minimum values of f subject to
g(x, y) = k is the following:

Answer 33

1. Find all x, y, λ such that ∇f = λ∇g and g(x, y) = k.
2. Evaluate f at these points (x, y) and choose the smallest and largest values.
   The number λ is called a Lagrange multiplier.

Question 34

Equation of the tangent plane to the level surface F (x, y, z) = k at P (x0, y0, z0)
is

Answer 34

Fx (x0, y0, z0) (x − x0) + Fy (x0, y0, z0) (y − y0) + Fz (x0, y0, z0) (z − z0) = 0

Question 35

Equation of the normal line at P (x0, y0, z0) is

Answer 35

(x − x0) / (Fx (x0, y0, z0)) = (y − y0) / (Fy (x0, y0, z0)) = (z − z0) / (Fz (x0, y0, z0)

Question 36

Line Integral

Answer 36

int f (x(t), y(t))|r′(t)| dt

Question 37

Line Integral in Space (3D)

Answer 37

int f(x(t), y(t), z(t)) √(dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2 dt

Question 38

Work Done

Answer 38

int F(r (t)) * |r′ (t)| dt

Question 39

Fundamental Theorem for Line Integrals

Answer 39

int ∇f · dr = f (r(b)) − f (r(a)).

Question 40

F is a conservative vector field if there is a function f such that

Answer 40

F = ∇f

Question 41

Green’s Theorem

Answer 41

double int (∂Q/∂x) − (∂P/∂y) dA.

Question 42

Curl

Answer 42

curlF = ∇F × F

Question 43

Vector Field F is conservative if

Answer 43

curl