Vector Calculus Flashcards
Distance Formula for (x,y,z)
PQ = ( (x1 - x2)2 + (y1 - y2)2 + (z1 - z2)2 )1/2
magnitude of v
||v|| = ( a2 + b2 + c2 )1/2
vector components
P = point (a,b,c)
Q = point (d,e,f)
PQ = < d - a , e - b , f - c>
midpoint formula
(a,b,c) + 1/2 * PQ
or
< (a1 + 2)/2 , (b1 +b2)/2 , (c1 + c2)/2 >
dot product
v * w = a1*a2 + b1*b2 + c1*c2
v * w = ||v|| * ||w|| * cos Ø
v * v = ||v||2
unit vector = ev
v / ||v||
vector projection of u onto v
u||
u|| = ( (u * v) / (||v||2) ) * v
u|| = (u * ev) * ev
u|| = ( (u * v) / (v * v) ) * v
parallel vectors
if lines through w and v are parrallel
or
w = Cv
for scalar C not equal to 0
equation of a sphere
of radius R
centered at (a, b, c)
(x - a)2 + (y - b)2 + (z - c)2 = R2
equation of a cylinder
of radius R
whose center line is the vertical axis through (a, b, 0)
(x - a)2 + (y - b)2 = R2
magnitude of projection
of u along v
||u|||| = ||u|| * cos Ø
Cross Product
v X w
orthogonal to v and w
v X w = - (v X w)
v X v = 0
v X w = 0 iff w = Cv or v = 0
length of v X w
||v X w|| = ||v|| * ||w|| * sinØ
||v X w||2 = ||v||2 * ||w||2 - (v * w)2
scalar projection of u onto v
or
the component of u along v
u * ev
where ev = v / ||v||
a vector v is orthogonal to w
if and only if
v * w = 0
the angle Ø between v and w is obtuse/acute
v * w < 0 is obtuse
v * w > 0 is acute
the decomposition of u with respect to v
u = u|| + uperp
where uperp is orthogonal to v
C * v X w = v X (C * w) =
= C (v X w)
(u + v) X w =
= u X w + v X w
translation
a vector v undergoes a translation when it is moved parallel to itself
without changing length or direction
w is the translate
equivalent
v and w are equivalent if w is a translate of v
linear combination
(if v and w are not parallel)
u = r * v + s * w
parallelogram spanned
parallelogram whose vertices are the origin and the terminal
points of v, w, and w + v
consists of linear combination
where 0
The line L through P0 = (x0 , y0 , z0) in the direction
of (parallel to) **v; ** where O is origin
vector parametrization :
parametric equation:
r(t) = OP0 + v = < x0 , y0 , z0> + t *
x = x0 + at ; y = y0 + bt ; z = z0 + zt
The line through the points P = (a 1 , b1 , c1) and Q = (a2 , b2 , c2)
vector parametrization:
parametric equation:
r(t) = (1 - t) * OP + t * OQ
x = a1 + (a2 - a1) * t ; y = b1 + (b2 - b1) * t ; z = c1 + (c2 - c1) * t
Equation of a line through origin that consists of the multiples
of the non-zero vector v =
vector parametrization:
parametric equation:
r0 = t * v =
x = a * t ; y = b * t ; z = c * t
unit vector cross products
i X j = k
**j ** X k = i
k X i = j
i X i = j X j = k X k = 0
parallelogram spanned by v and w
has area:
||v X w||
||v|| * ||w|| * sin Ø
parallelepiped spanned by u , v , and w
has volume
||v X w|| * ||u|| * |cos Ø|
u * (v X w) |
vector triple product
u * (v X w)
= the determinant of [u v w]
||v X w||2
||v||2 * ||w||2 - (v * w)2
right hand rule
fingers curl from x-axis to y-axis
thumb is in the positive z direction
coordinate planes
defined by setting one the
coordinates equal to 0
plane in R3
defined by
a*x + b*y + c*z = d
normal vector
n
a non-zero vector orthogonal to a point
P0 = (x0 , y0 , z0) determining
a plane passing through a point P0
P lies on a plane if
n * P0P = 0
or
n * OP = n * OP0
equation of a plane
through point P0 = (x0 , y0 , z0)
where normal vector = n = < a , b , c >
vector form:
n * < x , y , z > = d
scalar form:
a*(x - x0) + b*(y - y0) + c*(z - z0) = 0
parallel planes
the planes are parallel if they have a common normal vector
obtained by choosing normal vector n and varying
constant d in equation
a*x + b*y + c*z = d
collinear
points that lie on the same line
three points determine a plan if they are
not collinear
trace
the intersection of a plane with a coordinate plane
or a plane parallel to a coordinate plane
trace is a line unless the plane is parrallel to coordinate plane
(then trace is empty or is the plane itself)
obtained by setting x , y , or z = 0
ellipsoids
(x / a)2 + (y / b)2 + (z / c)2 = 1
hyperboloids of one sheet
(x / a)2 + (y / b)2 - (z / c)2 = 1
hyperboloids of two sheets
(x / a)2 + (y / b)2 - (z / c)2 = -1
elliptic cone
(x / a)2 + (y / b)2 = (z / c)2
elliptic paraboloid
z = (x / a)2 + (y / b)2
hyperbolic paraboloid
z = (x / a)2 - (y / b)2
orthogonal projection of vector u obtained from a vector v
u - u|| = u<strong>perp</strong>
PQ = ( (x1 - x2)2 + (y1 - y2)2 + (z1 - z2)2 )1/2
Distance Formula for (x,y,z)
||v|| = ( a2 + b2 + c2 )1/2
magnitude of v
PQ = < d - a , e - b , f - c>
vector components
P = point (a,b,c)
Q = point (d,e,f)
(a,b,c) + 1/2 * PQ
or
midpoint formula
v * w = a1*a2 + b1*b2 + c1*c2
v * w = ||v|| * ||w|| * cos Ø
v * v = ||v||2
dot product
v / ||v||
unit vector = ev
u|| = ( (u * v) / (||v||2) ) * v
u|| = (u * ev) * ev
u|| = ( (u * v) / (v * v) ) * v
vector projection of u onto v
u||
if lines through w and v are parrallel
or
w = Cv
for scalar C not equal to 0
parallel vectors
(x - a)2 + (y - b)2 + (z - c)2 = R2
equation of a sphere
of radius R
centered at (a, b, c)
(x - a)2 + (y - b)2 = R2
equation of a cylinder
of radius R
whose center line is the vertical axis through (a, b, 0)
||u|||| = ||u|| * cos Ø
magnitude of projection
of u along v
orthogonal to v and w
v X w = - (v X w)
v X v = 0
v X w = 0 iff w = Cv or v = 0
Cross Product
v X w
||v X w|| = ||v|| * ||w|| * sinØ
||v X w||2 = ||v||2 * ||w||2 - (v * w)2
length of v X w
u * ev
where ev = v / ||v||
scalar projection of u onto v
or
the component of u along v
if and only if
v * w = 0
a vector v is orthogonal to w
v * w < 0 is obtuse
v * w > 0 is acute
the angle Ø between v and w is obtuse/acute
u = u|| + uperp
where uperp is orthogonal to v
the decomposition of u with respect to v
= C (v X w)
C * v X w = v X (C * w) =
= u X w + v X w
(u + v) X w =
a vector v undergoes a translation when it is moved parallel to itself
without changing length or direction
w is the translate
translation
v and w are equivalent if w is a translate of v
equivalent
u = r * v + s * w
linear combination
(if v and w are not parallel)
parallelogram whose vertices are the origin and the terminal
points of v, w, and w + v
consists of linear combination
where 0
parallelogram spanned
r(t) = OP0 + v = < x0 , y0 , z0> + t *
x = x0 + at ; y = y0 + bt ; z = z0 + zt
The line L through P0 = (x0 , y0 , z0) in the direction
of (parallel to) **v; ** where O is origin
vector parametrization :
parametric equation:
r(t) = (1 - t) * OP + t * OQ
x = a1 + (a2 - a1) * t ; y = b1 + (b2 - b1) * t ; z = c1 + (c2 - c1) * t
The line through the points P = (a 1 , b1 , c1) and Q = (a2 , b2 , c2)
vector parametrization:
parametric equation:
r0 = t * v =
x = a * t ; y = b * t ; z = c * t
Equation of a line through origin that consists of the multiples
of the non-zero vector v =
vector parametrization:
parametric equation:
i X j = k
**j ** X k = i
k X i = j
i X i = j X j = k X k = 0
unit vector cross products
||v X w||
||v|| * ||w|| * sin Ø
parallelogram spanned by v and w
has area:
||v X w|| * ||u|| * |cos Ø|
u * (v X w) |
parallelepiped spanned by u , v , and w
has volume
u * (v X w)
= the determinant of [u v w]
vector triple product
||v||2 * ||w||2 - (v * w)2
||v X w||2
fingers curl from x-axis to y-axis
thumb is in the positive z direction
right hand rule
defined by setting one the
coordinates equal to 0
coordinate planes
defined by
a*x + b*y + c*z = d
plane in R3
a non-zero vector orthogonal to a point
P0 = (x0 , y0 , z0) determining
a plane passing through a point P0
normal vector
n
n * P0P = 0
or
n * OP = n * OP0
P lies on a plane if
vector form:
n * < x , y , z > = d
scalar form:
a*(x - x0) + b*(y - y0) + c*(z - z0) = 0
equation of a plane
through point P0 = (x0 , y0 , z0)
where normal vector = n = < a , b , c >
the planes are parallel if they have a common normal vector
obtained by choosing normal vector n and varying
constant d in equation
a*x + b*y + c*z = d
parallel planes
points that lie on the same line
three points determine a plan if they are
not collinear
collinear
the intersection of a plane with a coordinate plane
or a plane parallel to a coordinate plane
trace is a line unless the plane is parrallel to coordinate plane
(then trace is empty or is the plane itself)
obtained by setting x , y , or z = 0
trace
(x / a)2 + (y / b)2 + (z / c)2 = 1
ellipsoids
(x / a)2 + (y / b)2 - (z / c)2 = 1
hyperboloids of one sheet
(x / a)2 + (y / b)2 - (z / c)2 = -1
hyperboloids of two sheets
(x / a)2 + (y / b)2 = (z / c)2
elliptic cone
z = (x / a)2 + (y / b)2
elliptic paraboloid
z = (x / a)2 - (y / b)2
hyperbolic paraboloid
u - u|| = u<strong>perp</strong>
orthogonal projection of vector u obtained from a vector v
