Test 1: 11.1 to 12.3

Question 1

How do you find a unit vector?

Answer 1

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

Question 2

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

Answer 2

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

Question 3

What is the standard equation of a sphere?

Answer 3

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

Question 4

What is the midpoint formula in 3 dimensions/

Answer 4

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

Question 5

How do you know vectors are parallel?

Answer 5

If there is some scalar c such that u = cv

Question 6

How do you find the dot product?

Answer 6

u1v1 + u2v2+ u3v3

Question 7

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

Answer 7

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

Question 8

How do you determine if two vectors are orthogonal?

Answer 8

Their dot product is 0

Question 9

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

Answer 9

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

Question 10

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

Answer 10

1

Question 11

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

Answer 11

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

Question 12

What is the cross product of two vectors?

Answer 12

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

Question 13

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

Answer 13

Use the cross product.

Question 14

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

Answer 14

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

Question 15

What does it mean if u x v is 0?

Answer 15

u and v are scalar multiples of each other

Question 16

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

Answer 16

|| u x v ||

Question 17

What is the triple scalar product?

Answer 17

u dot (v x w)

Question 18

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

Answer 18

Take the absolute value of the triple scalar product

Question 19

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

Answer 19

a

Question 20

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

Answer 20

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

Question 21

What are the parametric equations for a line in space?

Answer 21

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

Question 22

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

Answer 22

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

Question 23

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

Answer 23

x1

Question 24

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

Answer 24

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

Question 25

How do you find the angle between two planes?

Answer 25

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

Question 26

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

Answer 26

||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.

Question 27

How do you find the distance between two parallel planes?

Answer 27

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.

Question 28

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

Answer 28

||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

Question 29

What is a cylinder?

Answer 29

A curve in a plane

Question 30

What is the equation of a cylinder?

Answer 30

x^2 + y^2 = a^2

Question 31

What is the equation of a quadric surface in space?

Answer 31

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

Question 32

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

Answer 32

Ellipsoid

Question 33

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

Answer 33

Hyperboloid of One Sheet

Question 34

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

Answer 34

Hyperboloid of Two Sheets

Question 35

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

Answer 35

Elliptic Cone

Question 36

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

Answer 36

Elliptic Paraboloid

Question 37

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

Answer 37

Hyperbolic Paraboloid

Question 38

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

Answer 38

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

Question 39

What are the equations for conversion to cylindrical coordinates?

Answer 39

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

Question 40

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

Answer 40

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)

Question 41

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

Answer 41

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

Question 42

What are the equations for conversion from rectangular to spherical?

Answer 42

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

Question 43

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

Answer 43

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

Question 44

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

Answer 44

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

Question 45

What does a vector valued function look like?

Answer 45

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

Question 46

How is velocity defined in terms of vectors?

Answer 46

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

Question 47

How is acceleration defined in terms of vectors?

Answer 47

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

Question 48

How is speed determined in terms of vectors?

Answer 48

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