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Calculus 3 > Test 1: 11.1 to 12.3 > Flashcards

Test 1: 11.1 to 12.3 Flashcards

1
Q

How do you find a unit vector?

A

v / ||v|| or 1 / ||v|| * v

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2
Q

How do you find the X and Y components of any nonzero vector v making an angle theta with the positive x axis?

A

||v|| or ||v|| cos theta*i + ||v|| sin theta * j

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3
Q

What is the standard equation of a sphere?

A

(x-x0)^2 + (y-y0)^2 + (z-z0)^2 = r^2

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4
Q

What is the midpoint formula in 3 dimensions/

A

(x1 + x2)/2, (y1+y2)/2, (z1+z2)/2

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5
Q

How do you know vectors are parallel?

A

If there is some scalar c such that u = cv

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6
Q

How do you find the dot product?

A

u1v1 + u2v2+ u3v3

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7
Q

How do you find the angle between two nonzero vectors u and v?

A

cos theta = u dot v / ||u|| ||v||

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8
Q

How do you determine if two vectors are orthogonal?

A

Their dot product is 0

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9
Q

How do you find the 3 angles between a nonzero vector and the three unit vectors i j k?

A

Use directions cosines cos alpha = v1 / ||v||, cos beta = v2 / ||v||, cos gamma = v3 / ||v||

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10
Q

What is the sum of the squares of the direction cosines?

A

1

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11
Q

What is the formula for projection of u onto v using the dot product(

A

Both vectors must be nonzero. projv^u = ( u dot v / ||v||^2) * v

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12
Q

What is the cross product of two vectors?

A

(u2v3 - u3v2)i - (u1v3 - u3v1)j + (u1v2 - u2v1)k

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13
Q

How do you get a vector orthogonal to two other vectors?

A

Use the cross product.

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14
Q

What’s a way to find the angle between u and v using the cross product?

A

|| u x v || = ||u|| ||v|| sin theta

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15
Q

What does it mean if u x v is 0?

A

u and v are scalar multiples of each other

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16
Q

What is the area of a parallelogram having u and v as adjacent sides?

A

|| u x v ||

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17
Q

What is the triple scalar product?

A

u dot (v x w)

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18
Q

How do you find the volume V of a parallelepiped with vectors u v and w as adjacent edges?

A

Take the absolute value of the triple scalar product

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19
Q

In the parametric equation x = x1 + at, which piece is the direction vector?

A

a

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20
Q

How are the symmetric equations of a line in space determined?

A

By eliminating the parameter t and obtaining x-x1/a = y-y1/b = z-z1/c

21
Q

What are the parametric equations for a line in space?

A

x=x1 + at, y=y1 + bt, z=z1+ct

22
Q

How can you find the point of intersection of two lines / planes in 3d space?

A

Solve the system of plane equations for all variables, plug in the solutions to get the other points.

23
Q

In the parametric equation x = x1 + at, which piece is the point the line passes through/

A

x1

24
Q

What is the standard equation of a plane containing the point (x1,y1,z1) and having normal vector n = ?

A

a(x-x1) + b(y-y1) + c(z-z1) = 0, or ax + by + cz = 0

25
Q

How do you find the angle between two planes?

A

Take the normal vectors of both planes (n1, n2): cos theta = |n1 dot n2| / ||n1|| ||n2||

26
Q

How do you find the distance between a point and a plane?

A

||projn^PQ|| = |PQ dot n| / ||n|| where Q is the point not in the plane, P is a point in the plane and n is normal to the plane.

27
Q

How do you find the distance between two parallel planes?

A

use D = |ax0 + by0 + cz0 + d| / sqrt(a^2 + b^2 + c^2) and plug in a point on one of the planes for x0,y0,z0. Then determine the direction vectors (a, b, c) from the coefficients on the equation of the second planet. Then plug in.

28
Q

What is the distance between a point and a line in space?

A

||PQ x u|| / ||u|| where Q is the point specified, u is a direction vector for the line (from the parametric equations) and P is a point on the line

29
Q

What is a cylinder?

A

A curve in a plane

30
Q

What is the equation of a cylinder?

A

x^2 + y^2 = a^2

31
Q

What is the equation of a quadric surface in space?

A

Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0

32
Q

What shape is the equation form x^2/a^2 + y^2/b^2 + z^2/c^2 = 1?

A

Ellipsoid

33
Q

What shape is the equation form x^2/a^2 + y^2/b^2 - z^2/c^2 = 1?

A

Hyperboloid of One Sheet

34
Q

What shape is the equation form x^2/a^2 - y^2/b^2 - z^2/c^2 = 1?

A

Hyperboloid of Two Sheets

35
Q

What shape is the equation form x^2/a^2 + y^2/b^2 - z^2/c^2 = 0?

A

Elliptic Cone

36
Q

What shape is the equation form z = x^2/a^2 + y^2/b^2?

A

Elliptic Paraboloid

37
Q

What shape is the equation form z = y^2/b^2 - x^2/a^2

A

Hyperbolic Paraboloid

38
Q

How is a point P in space represented in cylindrical coordinates?

A

r, theta, z, where (r,theta) is a polar representation of the projection of P in the xy plane and z is the directed distance from (r,theta) to P

39
Q

What are the equations for conversion to cylindrical coordinates?

A

x = r cos theta, y = r sin theta, z = z

40
Q

How is a point P in space represented in spherical coordinates?

A

rho, theta, phi where rho is the distance between P and the origin (always positive), theta is the same angle used in cylindrical coordinates, and phi is the angle between the positive z-axis and the line from the origin to P (between 0 and pi)

41
Q

What are the equations for conversion from spherical to rectangular coordinates?

A

x = rho sin phi cos theta, y = rho sin phi sin theta, z = rho cos phi

42
Q

What are the equations for conversion from rectangular to spherical?

A

rho^2 = x^2 = y^2 + z^2, tan theta = y/x, phi = arccos(z/sqrt(x^2+y^2+z^2))

43
Q

What are the equations for conversion from spherical to cylindrical, where r is greater than or equal to 0?

A

r^2 = rho^2 sin^2 phi, theta = theta, z = rho cos phi

44
Q

What are the equations for conversion from spherical to cylindrical, where r is greater than or equal to 0?

A

rho = sqrt(r^2 + z^2), theta = theta, phi = arccos(z/sqrt(r^2 + z^2))

45
Q

What does a vector valued function look like?

A

r(t) = f(t)i = g(t)j + h(t)k

46
Q

How is velocity defined in terms of vectors?

A

v(t) = r’(t) = x’(t)i + y’(t)j + z’(t)k

47
Q

How is acceleration defined in terms of vectors?

A

a(t) = r’‘(t) = x’‘(t)i + y’‘(t)j + z’‘(t)k

48
Q

How is speed determined in terms of vectors?

A

||v(t)|| = ||r’(t)|| = sqrt((x’(t))^2 + (y’(t))^2)

Calculus 3 (4 decks)

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