Mutli - Chap 11

What is the distance formula 2D

Distance formula 3D

d = √[( x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]

If k is constant, then x=k represents a plane parallel to what plane

yz plane

If k is constant, then y=k represents a plane parallel to what plane

xz

If k is constant, then z=k represents a plane parallel to what plane

xy

What is the equation of a sphere

r^2 = (x-h)^2 + (y-k)^2 + (z-l)^2

equation of an ellipse

(x-h)^2/a^2 + (y-k)^2/b^2 = 1

b is ALWAYS greater than a

in the equation for an ellipse what is the center

(h,k)

In the equation for an ellipse what is the major axis and what is the minor axis

The major axis is b and the minor axis is a

How do you find the dot product of two vectors

You multiply the corresponding parts and then add the numbers you get all together

How do you find the magnitude of a vector

|a| = square root (x^2 + y^2 + z^2)

What is an alternate way to find dot product

|a|(|b|)cos(theta) if you have both magnitude and theta

how can you find the angle between two vectors

use the equation
dot product = |a|(|b|)cos(theta)
and solve for theta

If the dot product is zero what does that mean

The vectors are perpendicular

How do you know that vectors are parallel

They are scalar multiples

what is a direction cosine

cos(a) = dot product of vectors / |a|
- corresponds to the coefficient of the "i" in an equation like a=-2i + j + 5k

what is a scalar projection of b onto a

dot product of a and b/ ||a||

what is a scalar projection aka. component of b along a

You are putting b onto the same direction as vector a and finding its value
"how much of vector b is pointed in the same direction as vector a"

What is vector projection

you are projecting a vector onto another vector for its direction only (the mag of the second vector doesn't matter)
- magnitude of comp b along a with direction

What is the formula for the comp b along a

||b||cos(theta)
- cuz of right triangle

what is the formula for projection of u onto v

(u⋅v)/(||v||^2)^* V

If you have the projection of u onto v (W1), how do you find the vector component of u orthogonal to v

W2 = U - W1
- since it is like the vector addition thing "tip to tail" so when you subtract one vector from the total vector you get the individual parts

If the component of one vector along the direction of another is zero what can you conclude about these two vectors?

the vectors are orthogonal

under what circumstances is comp b onto a = comp a onto b

If the dot product is not zero, then that means the magnitude of a and b are the same, so the vector a and b are the same length

what is the cosine, sine identity

cos^2 = 1 - sin^2

what is the sec, tan identity in terms of tan

sec^2 = 1 + tan^2

what is the sec, tan identity in terms of sec

tan^2 = sec^2 - 1

when you are left with sec(theta)d(theta) in the integral, what do you do

you multiply by (sec + tan)/(sec + tan)

when you are left with sec(theta) in the integral what is the shortcut

It ends up as ln|sec + tan| + C but in terms of theta so have to change it to x

when you are left with csc(theta)d(theta) in the integral, what do you do

you multiply by (csc + cot)/(csc + cot)

when you are left with csc(theta) in the integral what is the shortcut

It ends up as -ln|csc + cot| + C but in terms of theta so have to change it to x
- NEGATIVE ln
- make sure you do not forget

what should you do if you have something like x+3 in the numerator of the trig sub

split it into two integral, chances are you can do u sub for the x integral

how do you do cross product

Its the table thing where you do the cross of the column you're not in

if dot product is 0

the vectors are orthogonal

How do you find the area of a parallelogram with two vectors

||a x b||

is the output of cross product scalar or vector

it is a vector

how do you find the volume of a parallelepiped

the triple scalar product
u⋅(v x w)

how do you do triple scalar product

just watch the organic vid again for a reminder

What do you need to make an equation of a 3D line

a point and a vector (since it gives direction)

What is the vector equation of a line

r = r(0) + tv
kind of like y = mx+b

What are the parametric equations of a line

x = x0 +at
y= y0 + bt
z = z0 + ct

What are the symmetric equations of a line

(x-x0)/a = (y-y0)/b = (z-z0)/c
(made if you solve for t)

If a, b, or c equals zero on the symmetric equations of a line what does that mean

the particle is not changing positions in the direction

If you are given a point and a vector, how do you find the vector equation

you use the formula r = r0 +at
r0 is the point
a is the given vector
t is the variable

How do you find the parametric equations for a line that passes through points A and B if the points are given

You find the "slope" between the points and essentially find the vector between them. The values in that vector <a,b,c> are the coefficients in the parametric equation
x = point + at
y = point + bt
z = point + ct

how do you find when the line intersects the yz-plane when given the two points the line passes through

1) find the parametric equations
2) you set x = 0 since that's where it'll intersect the yz plane and solve for t
3) plug that t value into the other parametric equations and find the other values for y and z

how do you know if two lines are parallel based off of their parametric equations

you take the coefficient values of the parametric equations and make them a vector. compare the two. if they are scalar multiples, then they are parallel

What are skew lines

Lines that do not intersect and are not coplanar
- they could go through the same point but NOT at the same time

how do you determine if two lines are skew lines given the symmetric equations

1) you check that the lines aren't parallel by turning them into parametric, then vectors, then compare them to see if they're multiples
2) check that they don't intersect by setting the parametrics equal to each other (make one variable t and the other c). solve for t and c. If they work out then the lines intersect

What do you need to define a plane is space

You need a vector that is perpendicular to the plane since it restricts the direction the plane could possibly be in
- parallel vector would be infinite possibilities

What is a normal vector

a vector that is orthogonal to every vector in a given plane

how do you prove vectors are orthogonal

if the dot product is zero

what is the formula for the scalar equation of a plane given a point and normal vector <a,b,c>

a(x-x1) +b(y-y1) +c(z-z1) = 0

What is the linear equation of a plane

ax + by + cz + d = 0

How do you find the normal vector from the linear equation of a plane

vector = <a,b,c> based off of the equation ax + by + cz + d = 0

how do you find the equation of a plane that passes through three points

1) make 2 vectors from the points
2) do the cross product of the two vectors to find the orthogonal/normal vector
3) plug the values from the normal vector into the equation of a plane using one of the given points

How do you know if two planes are parallel

Their normal vectors are parallel

How do you know if two planes are perpendicular

the dot product of their normal vectors is zero

If the two planes are not parallel, then how do they intersect and what is the angle

They intersect in a straight line and the angle is called the acute angle between two vectors

How can you find the distance between a point and a plane

You create a vector between two known points, then use that vector to determine how much of that vector is made up by the orthogonal distance to the plane

What is the formula to find the distance between a point and a plane

comp vector (v) along normal vector (n) = (dot product v and n)/||n||

How do you find the distance between two planes

You find a vector between the two known points and use the normal vector of the plane to one of the points to do a comp of the vector onto the normal vector
- it gives you how much of that vector is between the two planes in the direction of the normal vector

how do you find the distance between two plane given the equation of the parallel planes

1) Find a point on both planes by setting y and z to zero and solving for x
2) Find the vector between the two points
3) comp of that vector along the normal vector

What are cylindrical coordinates

Its like a polar coordinate so its in (r, theta, z)
- z just stays the same

What is x equal to in polar

x = r*cos(theta)

What is y equal to in polar

y = r*sin(theta)

What is the x and y identity in polar

r^2 = x^2 + y^2

What is the tan identity in polar

tan(theta) = y/x

What are spherical coordinates

Coordinates in (rho, theta, phi)

What does rho represent

The distance from point p to the origin
- hypotenuse of triangle
- magnitude of vector to point p

What does phi represent

The angle between rho and the z-axis

What is rho equal to in cartesian

distance formula root( x + y + z) but the vars are squared

what does r equal in spherical coordinates

r = rho*sin(phi)