Calc 3 Midterm 1

Distance in R3

Convert from cylindrical coordinates to xyz coordinates

x=

y=

z=

rcosΘ

rsinΘ

z

Convert from xyz coordinates to cylindrical coordinates

r=

Θ=

z=

To convert from cylindrical coordinates to spherical coordinates:

p=

Θ=

Φ=

treat it like the relation btwn xyz and cylindrical but z=x and r=y

To convert from spherical coordinates to cylindrical coordinates:

r=

Θ=

z=

In R3

• circles in R2 become ___, ___ if ≤

• x=3 and y=2 would represent a

circular cylinder, solid circular cyinder

line

describing cylinders in R3

this is a (solid) circular cylinder of radius __ centered along the line/axis __, extending along the __ axis

given vectors a: ⟨ a1, a2,a3 ⟩ and b: ⟨ b1, b2,b3 ⟩

two formulas

Properties of dot product:

1. v⋅w

2. v⋅(u+w)

3. (cv)⋅w

4. 0 ⋅w=

5. v⋅v = ____ thus |v| = _____

w⋅v

v⋅u + v⋅w

c(v⋅w) = v⋅ (cw)

0

|v|² (≥0 for all v), √(v⋅v)

Cross product: u x v

if you have component form:

it gives a ____

vector

Area of parallelogram and area of triangle given two vectors that go from the same point

for triangle, divide it by 2

algebraic properties of the cross product: suppose u, v, and w are vectors in R3 and t is a scalar

1. v x u = ___ u x v (corollary: u x u = ___)

2. u x (v+w) = _____, (u+v) x w = _____

3. (tu) x v = t(u x v) = u x (tv)

4. u x 0 = ___

5. u⋅(v x w) = ____ (triple scalar product)

6. u x (v x w) = _______ (triple vector product)

-, 0

u x v + u x w, u x w + v x w

0

(u x v)⋅w

(u⋅w)v - (u⋅v)w

2 ways

Any vector c in the plane spanned by a and b must be

____ to ___.

orthogonal, axb

triple scalar product: volume of parallelepiped formed by

vectors u, v, w

V = _____

How to find the triple scalar product:

|u⋅(vxw)| note the brackets mean abs value

Do cross product v x w but replace i j k with u1 u2 u3

r = The displacement vector pointing from the axis of rotation (pivot point) to the point where the force is applied.

F = (Force Vector): The applied force vector causing the rotational tendency.

θ =angle between the vectors = ______

θ restriction

To determine a line in R3, we need to know:

_____ on the line P0 (x0, y0, z0) and a ___that’s ___ to the line〈a,b,c〉

• if u have 2 pts, the vector going thru them is parallel to line

vector equation of L: _____
parametric equations of L: ______

how to find other points on line: _____

one point, vector, parallel

plug in diff values of t

Parametric equation of a plane requires _____(x0, y0, z0) and

______ ⟨a1, b1, c1⟩ and ⟨a2, b2, c2⟩

Equation:_____

one point, two non-parallel vectors

Implicit equation of a plane

A plane is determined by ____ (x0, y0, z0) and a _____ (vector perp to plane) n = ⟨a, b, c⟩ .

Equation: ______, simplify to form Ax+By+Cz=D

a point, normal vector

Distance from point to line

• line is Q + tv, where Q = (x0,y0, z0)

• point is P

Distance =

Distance from point to plane

• plane is Ax+By+Cz=D, where n=⟨A, B, C⟩

• point is P

Distance =

Point C = any point on plane (set y=0 and z=0 and find x to make it easy)

Distance btwn 2 skew lines L1 and L2

• let v1 and v2 be directions of L1 and L2, then n= ____

• take a point P1 on L1 and P2 on L2

v1xv2

Work =

force • displacement

In standard spherical coordinates (ρ, θ, Φ)

ρ is the distance from the origin to the point.

Φ is the polar angle (or inclination) measured ____ from the _____

θ is the azimuthal angle measured in the xy plane ____ from the _____

downward, pos z axis

counterclockwise, positive x axis

In standard cylindrical coordinates (r, θ, Φ)

r is the distance from ____

θ is the azimuthal angle measured in the xy plane ____ from the _____

z is height above/below xy plane

z axis

counterclockwise, positive x axis

skew lines

not parallel and dont intersect

In R3, the number of planes containing two lines

• Identical lines:

• Parallel and distinct lines:

  • use the _____(which is the same for both) and create a 2nd vector by connecting ___ with a _____

• Intersecting lines (at a single point):

  • use _____ point and both ____

• Skew lines: 0 planes

• Infinitely many planes

• Exactly 1 plane, direction vector , a point on L1, a point on L2

• Exactly 1 plane, intersection, line direction vectors.

• 0 planes

how to find the point on the plane closest to a point

1. Find the ____ of plane

2. Construct the ____ passing through the ____ (parametric eqs). this is the shortest dist line

3. plug the parametric eqs into the ____ to find the intersection of this shortest dist line with the plane

4. plug ur value of ___ back into _____ to get the point

normal vector

perpendicular line, point

plane equation

t, parametric eqs

Closest point on a sphere to a plane

1. identify ____ and ___

2. find closest point on plane from center

3. Find the vector from the ____ to the ____ and turn it into ____

4. Closest point on sphere = ____

center and radius of sphere

center, plane's closest point, unit vector

Center + R (unit vector toward plane)

how to see if 2 lines intersect

• take their parametric equations, let one variable be __ and the other _

• set the x, y, and z components equal, see if there are values of t and s that work for all

t, s

Geometric meaning of setting coordinates constant:

ρ = c: A ____ of radius c centered at the origin.

r = c: A _____ of radius c centered around the ___.

Φ = c: A ___ opening up or down from the origin (or the flat xy-plane if Φ = π/2).

θ = c: A flat vertical ____ starting at the ___ and extending outward in that angular direction.

sphere

vertical cylinder, z-axis

cone

half-plane, z-axis

How to find Closest Point on a Line

Point is C and line is given by (x,y,z) = (x0,y0,z0)+t〈a,b,c〉

1. Write down a general point on the line by distributing t

2. Write the vector CP = P - C

3. The line’s direction vector is v. For the distance to be minimal, _____. solve for t

4. plug t back into P