Multivariable Calculus - Lecture #2

Dot product, Orthogonal Projection, Cross product,

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

Dot Product: 2 ways to find it

if you are given components:

if you are given angle between the 2 vectors:

it gives a ____

scalar

0 ≤θ ≤ π

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)

θ =angle between the vectors = ______

θ restriction

if θ = 0 or π, then the vectors are _____

if θ = π/2, then the vectors are _____

0 ≤θ ≤ π

parallel

orthogonal

How to prove parallel vectors

How to prove perpendicular (orthogonal vectors)

scalar multiples

dot product is 0

scalar/component projection 2 formulas:

It is positive for an ___ angle and negative for an ____ angle).

acute, obtuse

vector projection of v in the direction of u formula

the one youre projecting onto has the ____

magnitude * direction

scalar * vector

direction

direction cosines

cos²α+cos²β+cos²γ=___

1

how to find unit vector from direction cosines

Cross product: u x v

2 ways to find it

if you have component form:

if you know angle θ btwn u and v:

it gives a ____

|u||v|sinθ * unit vector perp to u and v

vector

u x v is a vector _____ to both u and v

____ is ALSO a vector orthogonal to u and v

u x v = ____(v x u)

orthogonal

v x u

-

if a⋅b = 0

if a x b = 0

a and b are orthogonal

a and b are parallel

|u x v| =

|u||v| sinθ

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

magnitude of cross product

magnitude of cross product / 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)

-

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

0

(u x v)⋅w

(u⋅w)v - (u⋅v)w

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

how to prove 3 vectors u, v, and w are coplanar

if u⋅(v x w) = 0

Scalar equation for planes in R3

need a ____ (x0, y0, z0) and a _____ ⟨a,b,c ⟩

standard form:

general/linear form:

point on plane, normal vector

A(x-x0)+B(y-y0)+C(z-z0) = 0

Ax + By + Cz = D

planes are parallel if______ they’re perp if_____

angle between planes = ____

their normal vector are parallel or same, their normals are perp

angle btwn normals

how to find plane equation from 3 points

• create two ____ from same ____

• do a _____ of those 2 vectors to get the ____

• fill out formula

vectors, point

cross product, normal vector

how to find plane equation from 1 point and a line in the plane

• from the line, you can get a ____ and a _____

• from the two points, get a _____, then do a _____ of those 2 vectors to get the normal

• fill out formula

point, vector

vector, cross product

how to find plane equation from 2 points and equation of a plane perp to it

• find vector on plane

• find ____ of other plane using its equation

• normal of the plane u wanna find is the cross product of the ____ and the _____

normal vector

vector on it, normal of the other plane

how to find angle between line and plane

• line gives you ____ plane gives you_____

• angle between = ______

vector, normal

90 - angle between the vector and normal

to find intersection of line and plane, “Plug” ___ into ____

line, plane

equation in line for R2 from point and normal vector

skew lines: lines that aren’t parallel but also do not ____

intersect

vector equation for plane

• n = a normal vector (perpendicular to the plane)

• r= position vector of a general point on the plane

• r0 = position vector of a specific point on the plane

n⋅(r−r0​)=0

or n⋅r = n⋅r0

find eq for line of intersection of 2 planes

• lines need point and a vector

• vector: _____ of _____ of the 2 planes

• point: set ______= _ for sys of eqs for the 2 planes

cross product, normals

x,y, or z = 0

distance btwn point and plane

distance btwn parallel planes

distance btwn skew lines

• skew lines can be contained by ______

• 1. find _______ of lines

• 2. find common normal: n = ______

• 3. lines have points, P1 & P2

• 4. take ____ and ______ to make eqs of planeS

• 5. find dist btwn the parallel planes

parallel planes

direction vectors v1 & v2

v1 x v2

P1 & n, P2 & n

dist btwn pt and line R3

|uxv| / |v|

dist btwn pt and line R2

-c if doing ax+by=c