Test 1: 11.1 to 12.3 Flashcards
How do you find a unit vector?
v / ||v|| or 1 / ||v|| * v
How do you find the X and Y components of any nonzero vector v making an angle theta with the positive x axis?
||v|| or ||v|| cos theta*i + ||v|| sin theta * j
What is the standard equation of a sphere?
(x-x0)^2 + (y-y0)^2 + (z-z0)^2 = r^2
What is the midpoint formula in 3 dimensions/
(x1 + x2)/2, (y1+y2)/2, (z1+z2)/2
How do you know vectors are parallel?
If there is some scalar c such that u = cv
How do you find the dot product?
u1v1 + u2v2+ u3v3
How do you find the angle between two nonzero vectors u and v?
cos theta = u dot v / ||u|| ||v||
How do you determine if two vectors are orthogonal?
Their dot product is 0
How do you find the 3 angles between a nonzero vector and the three unit vectors i j k?
Use directions cosines cos alpha = v1 / ||v||, cos beta = v2 / ||v||, cos gamma = v3 / ||v||
What is the sum of the squares of the direction cosines?
1
What is the formula for projection of u onto v using the dot product(
Both vectors must be nonzero. projv^u = ( u dot v / ||v||^2) * v
What is the cross product of two vectors?
(u2v3 - u3v2)i - (u1v3 - u3v1)j + (u1v2 - u2v1)k
How do you get a vector orthogonal to two other vectors?
Use the cross product.
What’s a way to find the angle between u and v using the cross product?
|| u x v || = ||u|| ||v|| sin theta
What does it mean if u x v is 0?
u and v are scalar multiples of each other
What is the area of a parallelogram having u and v as adjacent sides?
|| u x v ||
What is the triple scalar product?
u dot (v x w)
How do you find the volume V of a parallelepiped with vectors u v and w as adjacent edges?
Take the absolute value of the triple scalar product
In the parametric equation x = x1 + at, which piece is the direction vector?
a
How are the symmetric equations of a line in space determined?
By eliminating the parameter t and obtaining x-x1/a = y-y1/b = z-z1/c
What are the parametric equations for a line in space?
x=x1 + at, y=y1 + bt, z=z1+ct
How can you find the point of intersection of two lines / planes in 3d space?
Solve the system of plane equations for all variables, plug in the solutions to get the other points.
In the parametric equation x = x1 + at, which piece is the point the line passes through/
x1
What is the standard equation of a plane containing the point (x1,y1,z1) and having normal vector n = ?
a(x-x1) + b(y-y1) + c(z-z1) = 0, or ax + by + cz = 0
How do you find the angle between two planes?
Take the normal vectors of both planes (n1, n2): cos theta = |n1 dot n2| / ||n1|| ||n2||
How do you find the distance between a point and a plane?
||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.
How do you find the distance between two parallel planes?
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.
What is the distance between a point and a line in space?
||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
What is a cylinder?
A curve in a plane
What is the equation of a cylinder?
x^2 + y^2 = a^2
What is the equation of a quadric surface in space?
Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0
What shape is the equation form x^2/a^2 + y^2/b^2 + z^2/c^2 = 1?
Ellipsoid
What shape is the equation form x^2/a^2 + y^2/b^2 - z^2/c^2 = 1?
Hyperboloid of One Sheet
What shape is the equation form x^2/a^2 - y^2/b^2 - z^2/c^2 = 1?
Hyperboloid of Two Sheets
What shape is the equation form x^2/a^2 + y^2/b^2 - z^2/c^2 = 0?
Elliptic Cone
What shape is the equation form z = x^2/a^2 + y^2/b^2?
Elliptic Paraboloid
What shape is the equation form z = y^2/b^2 - x^2/a^2
Hyperbolic Paraboloid
How is a point P in space represented in cylindrical coordinates?
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
What are the equations for conversion to cylindrical coordinates?
x = r cos theta, y = r sin theta, z = z
How is a point P in space represented in spherical coordinates?
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)
What are the equations for conversion from spherical to rectangular coordinates?
x = rho sin phi cos theta, y = rho sin phi sin theta, z = rho cos phi
What are the equations for conversion from rectangular to spherical?
rho^2 = x^2 = y^2 + z^2, tan theta = y/x, phi = arccos(z/sqrt(x^2+y^2+z^2))
What are the equations for conversion from spherical to cylindrical, where r is greater than or equal to 0?
r^2 = rho^2 sin^2 phi, theta = theta, z = rho cos phi
What are the equations for conversion from spherical to cylindrical, where r is greater than or equal to 0?
rho = sqrt(r^2 + z^2), theta = theta, phi = arccos(z/sqrt(r^2 + z^2))
What does a vector valued function look like?
r(t) = f(t)i = g(t)j + h(t)k
How is velocity defined in terms of vectors?
v(t) = r’(t) = x’(t)i + y’(t)j + z’(t)k
How is acceleration defined in terms of vectors?
a(t) = r’‘(t) = x’‘(t)i + y’‘(t)j + z’‘(t)k
How is speed determined in terms of vectors?
||v(t)|| = ||r’(t)|| = sqrt((x’(t))^2 + (y’(t))^2)
