mat 175 memes

Question 1

hyperbola

Answer 1

(x-h)^2/a^2 - (y-k)^2/b^2 = 1. The minus is most important. Same pattern of stretching..? If a>b, line of symmetry is vertical; otherwise it is horizontal.

Question 2

u * u = (convert to magnittude)

Answer 2

||u||^2

Question 3

eillipsoid

Answer 3

Ellipsoid: x^2/a^2 + y^2/b^2 + z^2/c^2 = 1
Xyplane cross section: ellipse
Xz plane: ellipse
Yzplan: ellipse
Parallel to xyplane: ellipse, point, or empty set
Parallel to xzplane: ellipse, poimt, or empty set
Parallel to yzplane: ellipse, point, empty set

Question 4

cross product u x v

Answer 4

Always of 3d vectors. It’s the determinant of a 3x3 matrixwhose rows are <i>, u and v, in that order. To find each component of ther esulting vector, Take determinant of the part of the matrix that excludes the component’s corresponding column. Also, when finding the j component, substract it instead of adding it. </i>

Question 5

Find the distance between parallel planes

Answer 5

finding a point on a plane (x0, y0, z0) and using the formula. L = |Ax0 + By0+Cz0-D|/sqrt(A^2 + B^2 + C^2)

Question 6

standard form of equation for a plane

Answer 6

Ax+By+Cz=D or A(x-x1) + B(x-x1) + C(z-z1)=0 where <a> is the vector the plane is perpendicular to and (x1, y1, z1) is a point on the plane; simplifies to Ax+…</a>

Question 7

||u x v|| in terms of u and v

Answer 7

||u x v|| = ||u||||v||sin(angle between u and v)

Question 8

how to find parametric equations of a line from a direction vector and a point

Answer 8

take a direction vector ai + bj + ck and a point (x0, y0, z0) and the parametric equations of the line through that point are x = x0 + at, y = y0 + bt, z = z0 + ct.

Question 9

Hyperboloid of one sheet

Answer 9

x^2/a^2 + y^2/b^2 - z^2/c^2 = 1
Xyplane ellipse (messed up circle)
Xzplane hyperbola
Yzplane hyperbola

Looks like a torso!

Question 10

normal vector

Answer 10

perpendicular vector, orthogonal vector

Question 11

cylinder

Answer 11

Cylinders are the set of all points on lines parallel to l and that intersect C when l is a line not in a plane of C. Basically, along one dimension, it’s just a line.

Question 12

how to analyze weird shaped graphs

Answer 12

To analyze graphs, set dimenions to zero to identify traces on coordinate planes.
Name graphs by converting them into the appropriate form.

Question 13

line in vector notation

Answer 13

Line is determined by a fixed point P0 and a fixed direction vector v = ai + bj + ck for tehe line. It’s the set of all points P such that vector(P0P) is parallel to v ie vector(P0, P) = tv for all real t.

Question 14

||aU||

Answer 14

|a| ||U||

Question 15

projection of u on v, define and formulate

Answer 15

the part of u going in the v direction. prV(U) = ((uv)/||v||)(V/||V||); the magnitude of the part of u in the v direction times the direction of the unit vector. Simplified to ((uv)/||v||^2)(v).

Question 16

how to interprent contour plots

Answer 16

Contour plots are graphs along two dimensions at different points along the third dimension. Straightforward.

Question 17

how to sketch a plane from an equation like 3x-4y+22=24

Answer 17

Find intercepts, connect dots, shade.

Question 18

equation of a sphere

Answer 18

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

just the set of all points that are a particular distance from (h, k, l)

Question 19

key cross products of components

Answer 19

I x j = k, j x k = I, k x I = j (axes form right hand triples with one another)

Question 20

ellipse equation, graph

Answer 20

oval. (x-h)^2/a^2 + (y-k)^2/b^2 = 1. Larger a or b means stretching; h and k shift right/up

Question 21

symmetric equations

Answer 21

Solving each of the parametric equations for t and equating the results produces the symmetric equations for the line through (x0, y0, z0) with direction numbers a, b, c in the form:
x-x0/a = y-y0/b = z-z0/c.

Question 22

Hyperbolic paraboloid

Answer 22

z = y^2/b^2 - x^2/a^2
Xyplane: intersecting straight lines ???
Xzplane: parabola
Yzplane: parabola

weird multidimensional parabolas; turn an hyperboloid inside out or…?

Question 23

geometric relation of u x v with u and v

Answer 23

U x v is perpendicular to both u and v. The three form a righthand triple.

Question 24

perpendicularity criterion

Answer 24

Two vectors are perpendicular iff their dot product is 0.

Question 25

speed

Answer 25

||r’(t)|| magnitude of velocity

Question 26

u*v=(convert to magnitude)

Answer 26

u * v = ||u||||v||cos(smallest angle between u and v)

Question 27

how to describe a plane with vector language

Answer 27

Let n = <a> be a fixed nonzero vector and P1(x1, y1, z1) be a fixed point. The set of points P(x, y, z) satisfying vector(P1P) * n is the plane through P1 perpendicular to n.</a>

The Cartesian equation of the plane is acquired by writing vector(P1P) in component form = </a>

Question 28

how to find the equation of a plane through 3 points

Answer 28

Use the parallel observation of u x v = 0. Find two vectors connecting the three poitns (ex vector(P2P1), (P2P3). Their cross product is perpendicular is perpendicular to those vectors and so also to the plane containing them. Use standard form of plane to do the rest.

Question 29

parallelpiped

Answer 29

rectangular box!

Question 30

geogmetric meaning of u x v = 0

Answer 30

u and v are parallel

Question 31

circle

Answer 31

(x-h)^2 + (y-k)^2 = r^2. cENTER IS h,k, radius is r.

Question 32

U x (v x w) in terms of dot products

Answer 32

U x (v x w) = (uw)v - (uv)w

Question 33

higher partial derivatives

Answer 33

fxx is derivative by x twice. fyx is derivative of y and then x.

Question 34

(u x v) * w

Answer 34

u * (v x w)

Question 35

anticommutative law for cross products

Answer 35

U x v = -(v x u)

Question 36

ellipsoid

Answer 36

Ellipsoids, 
, x^2/a^2 + y^2/b^2 - z^2/c^2 = 1
Xyplane ellipse (messed up circle)
Xzplane hyperbola
Yzplane hyperbola

Question 37

Elliptic paraboloid

Answer 37

one sided parabola rotated along a dimension

0 = x^2/a^2 + y^2/b^2 - z
Xyplane: point
Xzplane: parabola
Yzplane: parabola

Question 38

partial derivative

Answer 38

often conveyed just by asking for fX(x0,y0) when asking for the partial derivative of f(x,y) with respect to x. Just treat the other variable(s) as a constant. Another way is like dz/dx or dz/dy since remember this is a function of two variables. Replace d with like a backwards 6.

Question 39

Hyerboloid of two sheets

Answer 39

Hyerboloid of two sheets x^2/a^2 - y^2
Xyplane: hyperbola
Xzplane: hyperbola
Yz plane: empty set

two-sided Parabola (ie a hyperbola) rotated along a dimension.

Question 40

U x u

Answer 40

0

Question 41

Distance L between a point (x0, y0, z0) and a plane Ax+By+Cz=D

Answer 41

L = |Ax0 + By0+Cz0-D|/sqrt(A^2 + B^2 + C^2)

Question 42

Elliptic cone

Answer 42

x^2/a^2 + y^2/b^2 - z^2/c^2 = 0
Xyplane: point
Xzplane: intersecting straight lines
Yzplane: intersecting straight line

hyperbola but like a cone.

Question 43

angle between two vectors

Answer 43

cos(angle) = u*v/||u||||v||. The dot product over the product of the two magnitudes.