12.1 Three-dimensional coordinate systems
(a, b, 0) is a projection onto the xy-plane, (0, b, c) is a projection onto the yz-plane, and (a, 0, c) is a projection
onto the xz-plane. p
The distance between point P1 (x1 , y1 , z1 ) and point P2 (x2 , y2 , z2 ) is given by |P1 P2 | = (x2 − x1 ) + (y2 − y1 ) + (z2 − z1 )
An equation of the sphere with center C(h, k, l) and radius r is given by (x − h)2 + (y − k)2 + (z − l)2 = r2
12.2 Vectors
If the displacement vector ⃗v has an initial point A and terminal point B, then ⃗v = AB. ⃗ Vectors with the
same length and direction are equal.
If ⃗u and ⃗v are vectors so that the initial point of ⃗v is the terminal point of ⃗u, then ⃗v + ⃗u represents the vector
from the initial point of ⃗u to the terminal point of ⃗v .
If c is a scalar and ⃗v is a vector, then c⃗v is the vector whose length is |c|||⃗v || and whose direction is either
the same or strictly opposite as the direction of ⃗v .
⃗ is ⃗a = ⟨x2 − x1 , y2 − y1 , z2 −
Given points A(x1 , y1 , z1 ) and B(x2 , y2 , z2 ), the vector ⃗a with representation AB
z1 ⟩. p
The length of ⃗a = ⟨a1 , a2 , a3 ⟩ is L(⃗a) = a21 + a22 + a23
Adding vectors: ⟨a1 , a2 , a3 ⟩ + ⟨b1 , b2 , b3 ⟩ = ⟨a1 + b1 , a2 + b2 , a3 + b3 ⟩
A unit vector is a vector whose length is one.
Properties of vectors:
• ⃗a + ⃗b = ⃗b + ⃗a • (cd)⃗a = c(d⃗a) • ⃗a + (−⃗a) = ⃗0
• ⃗a + ⃗0 = ⃗a • (c + d)⃗a = c⃗a + d⃗a
• c(⃗a + ⃗b) = c⃗a + c⃗b • ⃗a + (⃗b + ⃗c) = (⃗a + ⃗b) + ⃗c • 1⃗a = ⃗a
12.3 The dot product
If ⃗a = ⟨a1 , a2 , a3 ⟩ and ⃗b = ⟨b1 , b2 , b3 ⟩, then ⃗a · ⃗b = a1 b1 + a2 b2 + a3 b3 .
If θ is the angle between ⃗a and ⃗b, then ⃗a · ⃗b = |⃗a||⃗b| cos(θ).
Some properties of dot products:
• ⃗a · ⃗a = |⃗a|2 • ⃗a · (⃗b · ⃗c) = ⃗a · ⃗b + ⃗a · ⃗c • ⃗a · ⃗b = ⃗b · ⃗a
• ⃗0 · ⃗a = ⃗0 • (c⃗a) · ⃗b = c(⃗a · ⃗b) = ⃗a · (c⃗b)
Two vectors are orthogonal if and only if ⃗a · ⃗b = 0.
The direction angles of ⃗a are the angles α, β, γ that ⃗a makes with the positive x, y, z-axis, respectively. They
are given by:
• cos(α) = a1
|⃗
a| • cos(β) = a2
|⃗
a| • cos(γ) = a3
|⃗
a|
⃗
The scalar projection of ⃗b onto ⃗a: compa⃗b = ⃗a|⃗a·b| .
⃗
The vector projection of ⃗b onto ⃗a: proja⃗b = ⃗a·b2 ⃗a |⃗
a|
1
, 12.4 The cross product
⃗i ⃗j ⃗k
⃗ ⃗
If ⃗a = ⟨a1 , a2 , a3 ⟩ and b = ⟨b1 , b2 , b3 ⟩, then ⃗a × b = a1 a2 a3 = ⟨a2 b3 − a3 b2 , a3 b1 − a1 b3 , a1 b2 − a2 b1 ⟩
b1 b2 b3
The vector ⃗a × ⃗b is orthogonal to both ⃗a and ⃗b
|⃗a × ⃗b| = |⃗a||⃗b| sin θ where θ is the angle between ⃗a and ⃗b
Two nonzero vectors ⃗a and ⃗b are parallel if and only if ⃗a × ⃗b = ⃗0
The length of the cross product ⃗a × ⃗b is equal to the area of the parallelogram determined by ⃗a and ⃗b
The volume of the parallelpiped determined by the vectors ⃗a, ⃗b, ⃗c is: V = |⃗a · (⃗b × ⃗c)|
Properties of the cross product:
• ⃗a × ⃗b = −⃗b × ⃗a • ⃗a · (⃗b × ⃗c) = (⃗a × ⃗b) · ⃗c • (⃗a + ⃗b) × ⃗c = ⃗a × ⃗c + ⃗b × ⃗c
• ⃗a × (⃗b + ⃗c) = ⃗a × ⃗b + ⃗a × ⃗c • (c⃗a) × ⃗b = c(⃗a × ⃗b) = ⃗a × (c⃗b) • ⃗a ×(⃗b×⃗c) = (⃗a ·⃗c)·⃗b−(⃗a ·⃗b)·⃗c
12.5 Equations of lines and planes
Parametric equations for a line through the point (x0 , y0 , z0 ) and parallel to the direction vector ⟨a, b, c⟩ are
• x = x0 + at • y = y0 + bt • z = z0 + ct
Symmetric equations: x−x a
0
= y−y b
0
= z−z c
0
The line segment from r⃗0 to r⃗1 is given by the vector equation ⃗r(t) = (1 − t)r⃗0 + tr⃗1 , 0 ≤ t ≤ 1
Vector equation of a plane: ⃗n · ⃗r = ⃗n · r⃗0 .
Scalar equation of a plane through P0 (x0 , y0 , z0 ) with normal vector ⃗n = ⟨a, b, c⟩ is a(x − x0 ) + b(y − y0 ) +
c(z − z0 ) = 0
The distance D from the point P1 (x1 , y1 , z1 ) to the plane ax + by + cz + d = 0 is D = |ax√ 1 +by1 +cz1 +d|
a2 +b2 +c2
13.1 Vector functions and space curves
If ⃗r(t) = ⟨f (t), g(t), h(t)⟩, then limt→a ⃗r(t) = ⟨limt→a f (t), limt→a g(t), limt→a h(t)⟩ and ⃗r(t) is continuous at
a if limt→a ⃗r(t) = ⃗r(a)
The domain of a vector function is the union of the domain of its components.
13.2 Derivatives and integrals of vector functions
d⃗
r
dt = ⃗r′ (t) = limh→0 ⃗r(t+h)−⃗
h
r (t)
. Then ⃗r′ (t) is called the tangent vector.
If ⃗r(t) = ⟨f (t), g(t), h(t)⟩, then ⃗r′ (t) = ⟨f ′ (t), g ′ (t), h′ (t)⟩. Some rules:
• d
dt [⃗
u(t) + ⃗v (t)] = ⃗u′ (t) + ⃗v ′ (t) • d
dt [⃗
u(t) · ⃗v t] = ⃗u′ (t) · ⃗v (t) + ⃗u(t) · ⃗v ′ (t)
• d
dt [c⃗
u(t)] = c⃗u′ (t) • d
dt [⃗
u(t) × ⃗v t] = ⃗u′ (t) × ⃗v (t) + ⃗u(t) × ⃗v ′ (t)
• d
dt [f (t)⃗
u(t)] = f ′ (t)⃗u(t) + f (t)⃗u′ (t) • d
dt [⃗
u(f (t)))] = f ′ (t)⃗u′ (f (t))
A unit vector that has the same direction as the tangent vector is called the unit tangent vector and is
′
defined by T⃗ (t) = ||⃗⃗rr′ (t)||
(t)
If ||⃗r(t)|| = c (a constant), then ⃗r′ (t) is orthogonal to ⃗r(t) for all t
´ ´ ´ ´ ´b
⃗
If ⃗r(t) = ⟨f (t), g(t), h(t)⟩, then ⃗r(t)dt == ⟨ f (t)dt, g(t)dt, h(t)dt⟩ = R(t) ⃗
and a ⃗r(t)dt = R(b) ⃗
− R(a)
2
(a, b, 0) is a projection onto the xy-plane, (0, b, c) is a projection onto the yz-plane, and (a, 0, c) is a projection
onto the xz-plane. p
The distance between point P1 (x1 , y1 , z1 ) and point P2 (x2 , y2 , z2 ) is given by |P1 P2 | = (x2 − x1 ) + (y2 − y1 ) + (z2 − z1 )
An equation of the sphere with center C(h, k, l) and radius r is given by (x − h)2 + (y − k)2 + (z − l)2 = r2
12.2 Vectors
If the displacement vector ⃗v has an initial point A and terminal point B, then ⃗v = AB. ⃗ Vectors with the
same length and direction are equal.
If ⃗u and ⃗v are vectors so that the initial point of ⃗v is the terminal point of ⃗u, then ⃗v + ⃗u represents the vector
from the initial point of ⃗u to the terminal point of ⃗v .
If c is a scalar and ⃗v is a vector, then c⃗v is the vector whose length is |c|||⃗v || and whose direction is either
the same or strictly opposite as the direction of ⃗v .
⃗ is ⃗a = ⟨x2 − x1 , y2 − y1 , z2 −
Given points A(x1 , y1 , z1 ) and B(x2 , y2 , z2 ), the vector ⃗a with representation AB
z1 ⟩. p
The length of ⃗a = ⟨a1 , a2 , a3 ⟩ is L(⃗a) = a21 + a22 + a23
Adding vectors: ⟨a1 , a2 , a3 ⟩ + ⟨b1 , b2 , b3 ⟩ = ⟨a1 + b1 , a2 + b2 , a3 + b3 ⟩
A unit vector is a vector whose length is one.
Properties of vectors:
• ⃗a + ⃗b = ⃗b + ⃗a • (cd)⃗a = c(d⃗a) • ⃗a + (−⃗a) = ⃗0
• ⃗a + ⃗0 = ⃗a • (c + d)⃗a = c⃗a + d⃗a
• c(⃗a + ⃗b) = c⃗a + c⃗b • ⃗a + (⃗b + ⃗c) = (⃗a + ⃗b) + ⃗c • 1⃗a = ⃗a
12.3 The dot product
If ⃗a = ⟨a1 , a2 , a3 ⟩ and ⃗b = ⟨b1 , b2 , b3 ⟩, then ⃗a · ⃗b = a1 b1 + a2 b2 + a3 b3 .
If θ is the angle between ⃗a and ⃗b, then ⃗a · ⃗b = |⃗a||⃗b| cos(θ).
Some properties of dot products:
• ⃗a · ⃗a = |⃗a|2 • ⃗a · (⃗b · ⃗c) = ⃗a · ⃗b + ⃗a · ⃗c • ⃗a · ⃗b = ⃗b · ⃗a
• ⃗0 · ⃗a = ⃗0 • (c⃗a) · ⃗b = c(⃗a · ⃗b) = ⃗a · (c⃗b)
Two vectors are orthogonal if and only if ⃗a · ⃗b = 0.
The direction angles of ⃗a are the angles α, β, γ that ⃗a makes with the positive x, y, z-axis, respectively. They
are given by:
• cos(α) = a1
|⃗
a| • cos(β) = a2
|⃗
a| • cos(γ) = a3
|⃗
a|
⃗
The scalar projection of ⃗b onto ⃗a: compa⃗b = ⃗a|⃗a·b| .
⃗
The vector projection of ⃗b onto ⃗a: proja⃗b = ⃗a·b2 ⃗a |⃗
a|
1
, 12.4 The cross product
⃗i ⃗j ⃗k
⃗ ⃗
If ⃗a = ⟨a1 , a2 , a3 ⟩ and b = ⟨b1 , b2 , b3 ⟩, then ⃗a × b = a1 a2 a3 = ⟨a2 b3 − a3 b2 , a3 b1 − a1 b3 , a1 b2 − a2 b1 ⟩
b1 b2 b3
The vector ⃗a × ⃗b is orthogonal to both ⃗a and ⃗b
|⃗a × ⃗b| = |⃗a||⃗b| sin θ where θ is the angle between ⃗a and ⃗b
Two nonzero vectors ⃗a and ⃗b are parallel if and only if ⃗a × ⃗b = ⃗0
The length of the cross product ⃗a × ⃗b is equal to the area of the parallelogram determined by ⃗a and ⃗b
The volume of the parallelpiped determined by the vectors ⃗a, ⃗b, ⃗c is: V = |⃗a · (⃗b × ⃗c)|
Properties of the cross product:
• ⃗a × ⃗b = −⃗b × ⃗a • ⃗a · (⃗b × ⃗c) = (⃗a × ⃗b) · ⃗c • (⃗a + ⃗b) × ⃗c = ⃗a × ⃗c + ⃗b × ⃗c
• ⃗a × (⃗b + ⃗c) = ⃗a × ⃗b + ⃗a × ⃗c • (c⃗a) × ⃗b = c(⃗a × ⃗b) = ⃗a × (c⃗b) • ⃗a ×(⃗b×⃗c) = (⃗a ·⃗c)·⃗b−(⃗a ·⃗b)·⃗c
12.5 Equations of lines and planes
Parametric equations for a line through the point (x0 , y0 , z0 ) and parallel to the direction vector ⟨a, b, c⟩ are
• x = x0 + at • y = y0 + bt • z = z0 + ct
Symmetric equations: x−x a
0
= y−y b
0
= z−z c
0
The line segment from r⃗0 to r⃗1 is given by the vector equation ⃗r(t) = (1 − t)r⃗0 + tr⃗1 , 0 ≤ t ≤ 1
Vector equation of a plane: ⃗n · ⃗r = ⃗n · r⃗0 .
Scalar equation of a plane through P0 (x0 , y0 , z0 ) with normal vector ⃗n = ⟨a, b, c⟩ is a(x − x0 ) + b(y − y0 ) +
c(z − z0 ) = 0
The distance D from the point P1 (x1 , y1 , z1 ) to the plane ax + by + cz + d = 0 is D = |ax√ 1 +by1 +cz1 +d|
a2 +b2 +c2
13.1 Vector functions and space curves
If ⃗r(t) = ⟨f (t), g(t), h(t)⟩, then limt→a ⃗r(t) = ⟨limt→a f (t), limt→a g(t), limt→a h(t)⟩ and ⃗r(t) is continuous at
a if limt→a ⃗r(t) = ⃗r(a)
The domain of a vector function is the union of the domain of its components.
13.2 Derivatives and integrals of vector functions
d⃗
r
dt = ⃗r′ (t) = limh→0 ⃗r(t+h)−⃗
h
r (t)
. Then ⃗r′ (t) is called the tangent vector.
If ⃗r(t) = ⟨f (t), g(t), h(t)⟩, then ⃗r′ (t) = ⟨f ′ (t), g ′ (t), h′ (t)⟩. Some rules:
• d
dt [⃗
u(t) + ⃗v (t)] = ⃗u′ (t) + ⃗v ′ (t) • d
dt [⃗
u(t) · ⃗v t] = ⃗u′ (t) · ⃗v (t) + ⃗u(t) · ⃗v ′ (t)
• d
dt [c⃗
u(t)] = c⃗u′ (t) • d
dt [⃗
u(t) × ⃗v t] = ⃗u′ (t) × ⃗v (t) + ⃗u(t) × ⃗v ′ (t)
• d
dt [f (t)⃗
u(t)] = f ′ (t)⃗u(t) + f (t)⃗u′ (t) • d
dt [⃗
u(f (t)))] = f ′ (t)⃗u′ (f (t))
A unit vector that has the same direction as the tangent vector is called the unit tangent vector and is
′
defined by T⃗ (t) = ||⃗⃗rr′ (t)||
(t)
If ||⃗r(t)|| = c (a constant), then ⃗r′ (t) is orthogonal to ⃗r(t) for all t
´ ´ ´ ´ ´b
⃗
If ⃗r(t) = ⟨f (t), g(t), h(t)⟩, then ⃗r(t)dt == ⟨ f (t)dt, g(t)dt, h(t)dt⟩ = R(t) ⃗
and a ⃗r(t)dt = R(b) ⃗
− R(a)
2
