Calculus 2
February 2016
Chapter 12
A vector has a direction and length. We call position vector a vector that goes from the origin to a
~ is
point. If there are the points A(x1 , y1 , z1 ) and B(x2 , y2 , z2 ), the vector a with representation AB
a = hx2 − x1 , y2 − y1 , z2 − z1 i
p
The legth of a vector is determined doing: |a| = a21 + a22 . Standard basis vectors:
i = h1, 0, 0i j =h0, 1, 0i k = h0, 0, 1i
a =a1 i + a2 j + a3 k
a
The unit vector of a vector a 6= 0: u = |a|
The Dot Product
a·b
a · b = a1 b1 + a2 b2 + a3 b3 → a · b = |a||b| cos theta → cos θ =
|a||b|
Two vectors are orthogonal if and only if a·b = 0. If a and b point in the same direction a·b = |a||b|
and if a and b point in opposite directions a · b = −|a||b|.
Direction Angles and Direction Cosines
The direction angles of a nonzero vector a are the angles α, β, and γ that a makes with the positive
x-, y-, and z-axes. The direction cosines of the vector a are:
a1 a2 a3
cosα = cosβ = cosγ = → cos2 α + cos2 β + cos2 γ = 1
|a| |a| |a|
Projections
a
Scalar projection of b onto a: compa b = |a| b
a·b
Vector projection of b onto a: proja b = |a|2
a
The Cross Product
i j k
c = a × b ≡ a1 a2 a3 c is orthogonal to both a and b. |a × b| = |a||b| sin θ
b1 b2 b3
1
, If the cross product is equal to 0, then the two vectors are parallel to each other. For the standard
basis vectors i, j, and k we obtain:
i×j =k j×k =i k×i=j
j × i = −k k × j = −i i × k = −j
Some properties of the cross product:
a · (b × c) = (a × b) · c
a × (b × c) = b(ac) − c(ab) → Bac Cab Rule!
Triple Product
a1 a2 a3
a·(b×c) = b1 b2 b3 → It0 s the volume of the parallelepiped determined by the vectors a, b and c.
c1 c2 c3
Equations of lines and planes
A line L in three-dimensional space is determined when we know a point P on L and the direction
of L. The vector equation of L is:
r = r0 + tv (tv = a)
The parametric equations of the line L through the point P0 (x0 , y0 , z0 ) and parallel to the vector
v = ha, b, ci are:
x = x0 + at y = y0 + bt z = z0 + ct
The symmetric equations of L are:
x − x0 y − y0 z − z0
= =
a b c
The line segment from r0 to r1 is given by the vector equation:
r(t) = (1 − t)r0 + tr1 0≤t≤1
Planes
Space determined by a point P0 (x0 , y0 , z0 ) and a normal vector that is orthogonal to the plane.
Two planes are parallel if their normal vectors are parallel (they will be the same normal vector).
The angle between 2 planes is defined as the acute angle between their normal vectors.
a(x − x0 ) + b(y − y0 ) + c(z − z0 ) = 0 → ax + by + cz + d = 0
Cylinder and quadratic surfaces
In order to sketch the graph of a surface, it is useful to determine the curves of intersection of
the surface with planes parallel to the coordinate planes. These curves are called traces (or cross-
sections) of the surface.
A cylinder is a surface that consists of all lines (called rulings) that are parallel to a given line and
pass through a given plane curve. If one of the variables x,y,z is missing, then it’s a cylinder.
A quadratic surface is a second-degree equation in the three variables x, y, and z.
Ax2 + By 2 + Cz 2 + J = 0 Ax2 + By 2 + Iz = 0
Page 2
February 2016
Chapter 12
A vector has a direction and length. We call position vector a vector that goes from the origin to a
~ is
point. If there are the points A(x1 , y1 , z1 ) and B(x2 , y2 , z2 ), the vector a with representation AB
a = hx2 − x1 , y2 − y1 , z2 − z1 i
p
The legth of a vector is determined doing: |a| = a21 + a22 . Standard basis vectors:
i = h1, 0, 0i j =h0, 1, 0i k = h0, 0, 1i
a =a1 i + a2 j + a3 k
a
The unit vector of a vector a 6= 0: u = |a|
The Dot Product
a·b
a · b = a1 b1 + a2 b2 + a3 b3 → a · b = |a||b| cos theta → cos θ =
|a||b|
Two vectors are orthogonal if and only if a·b = 0. If a and b point in the same direction a·b = |a||b|
and if a and b point in opposite directions a · b = −|a||b|.
Direction Angles and Direction Cosines
The direction angles of a nonzero vector a are the angles α, β, and γ that a makes with the positive
x-, y-, and z-axes. The direction cosines of the vector a are:
a1 a2 a3
cosα = cosβ = cosγ = → cos2 α + cos2 β + cos2 γ = 1
|a| |a| |a|
Projections
a
Scalar projection of b onto a: compa b = |a| b
a·b
Vector projection of b onto a: proja b = |a|2
a
The Cross Product
i j k
c = a × b ≡ a1 a2 a3 c is orthogonal to both a and b. |a × b| = |a||b| sin θ
b1 b2 b3
1
, If the cross product is equal to 0, then the two vectors are parallel to each other. For the standard
basis vectors i, j, and k we obtain:
i×j =k j×k =i k×i=j
j × i = −k k × j = −i i × k = −j
Some properties of the cross product:
a · (b × c) = (a × b) · c
a × (b × c) = b(ac) − c(ab) → Bac Cab Rule!
Triple Product
a1 a2 a3
a·(b×c) = b1 b2 b3 → It0 s the volume of the parallelepiped determined by the vectors a, b and c.
c1 c2 c3
Equations of lines and planes
A line L in three-dimensional space is determined when we know a point P on L and the direction
of L. The vector equation of L is:
r = r0 + tv (tv = a)
The parametric equations of the line L through the point P0 (x0 , y0 , z0 ) and parallel to the vector
v = ha, b, ci are:
x = x0 + at y = y0 + bt z = z0 + ct
The symmetric equations of L are:
x − x0 y − y0 z − z0
= =
a b c
The line segment from r0 to r1 is given by the vector equation:
r(t) = (1 − t)r0 + tr1 0≤t≤1
Planes
Space determined by a point P0 (x0 , y0 , z0 ) and a normal vector that is orthogonal to the plane.
Two planes are parallel if their normal vectors are parallel (they will be the same normal vector).
The angle between 2 planes is defined as the acute angle between their normal vectors.
a(x − x0 ) + b(y − y0 ) + c(z − z0 ) = 0 → ax + by + cz + d = 0
Cylinder and quadratic surfaces
In order to sketch the graph of a surface, it is useful to determine the curves of intersection of
the surface with planes parallel to the coordinate planes. These curves are called traces (or cross-
sections) of the surface.
A cylinder is a surface that consists of all lines (called rulings) that are parallel to a given line and
pass through a given plane curve. If one of the variables x,y,z is missing, then it’s a cylinder.
A quadratic surface is a second-degree equation in the three variables x, y, and z.
Ax2 + By 2 + Cz 2 + J = 0 Ax2 + By 2 + Iz = 0
Page 2
