Dot
Product. The dot product or scalar product of two vectors is defined,
a
· b ≡ |a| |b| cos θ,
where
θ is the angle from a to b. From this definition one can derive the
following properties:
·
a
· b = b · a, commutative.
·
α(a
· b) = (αa) · b = a · (αb), associativity of scalar
multiplication.
·
a
· (b + c) = a · b + a · c, distributive. (See Exercise 2.1.)
·
eiej
= δij. In three dimensions, this is
i · i = j · j = k · k =
1, i · j = j · k = k ·
i = 0.
·
a
· b = aibi ≡ a1b1
+···+ anbn, dot product in terms of rectangular
components.
· If
a · b = 0 then either a and b are orthogonal, (perpendicular), or one of a
and b are zero.
The
Angle Between Two Vectors. We can use the dot
product to find the angle between two vectors, a and b. From the definition of
the dot product,
a
· b = |a| |b| cos θ.
If
the vectors are nonzero, then,
Parametric
Equation of a Line. Consider a line in Rn
that passes through the point a and is parallel to the vector t, (tangent). A
parametric equation of the line is
x
= a + ut, u ∈ R.
Implicit
Equation of a Line In 2D. Consider a
line in R2 that passes through the point a and is normal,
(orthogonal, perpendicular), to the vector n. All the lines that are normal to
n have the property that x · n is a constant, where x is any point on the line.
(See Figure 2.5.) x · n = 0 is the line that is
normal to n and passes through the origin. The line that is normal to n and
passes through the point a is
x
· n = a · n.
The
normal to a line determines an orientation of the line. The normal points in
the direction that is above the line. A point b is (above/on/below) the line if
(b − a)·n is (positive/zero/negative). The signed distance of a point
Figure
2.5: Equation for a line.
b from the line x · n = a · n is
Implicit
Equation of a Hyperplane. A hyperplane in Rn
is an n − 1 dimensional “sheet” which passes through a given point and is
normal to a given direction. In R3 we call this a plane. Consider a
hyperplane that passes through the point a and is normal to the vector n. All
the hyperplanes that are normal to n have the property that x · n is a
constant, where x is any point in the hyperplane. x · n = 0 is the hyperplane
that is normal to n and passes through the origin. The hyperplane that is
normal to n and passes through the point a is,
x
· n = a · n.
The
normal determines an orientation of the hyperplane. The normal points in the
direction that is above the hyperplane. A point b is (above/on/below) the
hyperplane if (b − a) · n is (positive/zero/negative). The signed distance of a
point b from the hyperplane x · n = a · n is,
Right
and Left-Handed Coordinate Systems.
Consider a rectangular coordinate system in two dimensions. Angles are measured
from the positive x axis in the direction of the positive y axis. There are two
ways of labeling the axes. (See Figure 2.6.) In
one the angle increases in the counterclockwise direction and in the other the
angle increases in the clockwise direction. The former is the familiar
Cartesian coordinate system.
Figure
2.6: There are two ways of labeling the axes in two dimensions.
There are also two ways of labeling the axes in a three-dimensional rectangular coordinate system.
These are called right-handed and left-handed coordinate
systems. See Figure 2.7. Any other labelling of
the axes could be rotated into one of these configurations. The right-handed
system is the one that is used by default. If you put your right thumb in the
direction of the z axis in a right-handed coordinate system, then your
fingers curl in the direction from the x axis to the y axis.
Cross
Product. The cross product or vector product is
defined,
a
× b = |a| |b| sin θ n,
where
θ is the angle from a to b and n is a unit vector that is orthogonal to
a and b and in the direction such that the ordered triple of vectors a, b and n
form a right-handed system.
Figure
2.7: Right and left handed coordinate systems.
You can visualize the direction of a × b by applying the right hand rule. Curl the fingers of your right hand in the direction from a to b. Your thumb points in the direction of a × b. Warning: Unless you are a lefty, get in the habit of putting down your pencil before applying the right hand rule.
The
dot and cross products behave a little differently. First note that unlike the
dot product, the cross product is not commutative. The magnitudes of a × b and
b × a are the same, but their directions are opposite. (See Figure 2.8.)
Let
a
× b = |a| |b| sin θ n and b × a = |b| |a| sin φ m.
The
angle from a to b is the same as the angle from b to a. Since {a, b, n} and {b,
a, m} are right-handed systems, m points in the opposite direction as n. Since
a × b = −b × a we say that the cross product is anti-commutative.
Next
we note that since
|a
× b| = |a| |b| sin θ,
the
magnitude of a × b is the area of the parallelogram defined by the two vectors.
(See Figure 2.9.) The area of the triangle
defined by two vectors is then ½ |a × b|.
From
the definition of the cross product, one can derive the following properties:
Figure
2.8: The cross product is anti-commutative.
a a
Figure 2.9: The parallelogram and the triangle defined by two vectors.
·
a
× b = −b × a, anti-commutative.
·
α(a
× b) = (αa) × b = a × (αb), associativity of scalar
multiplication.
·
a
× (b + c) = a × b + a × c, distributive.
·
(a
× b) × c ≠ a × (b × c). The cross product is not associative.
·
i
× i = j × j = k × k = 0.
·
i
× j = k, j × k = i, k × i = j.
· 
cross product in terms of
rectangular components.
·
If
a · b = 0 then either a and b are parallel or one of a or b is zero.
Scalar
Triple Product. Consider the volume of the
parallelopiped defined by three vectors. (See Figure 2.10.)
The area of the base is ||b| |c| sin θ|, where θ is the angle
between b and c. The height is |a| cos φ, where φ is the angle
between b × c and a. Thus the volume of the parallelopiped is |a| |b| |c| sin θ
cos φ.
Figure
2.10: The parallelopiped defined by three vectors.
Note that
|a
· (b × c)| = |a · (|b| |c| sin θ n)|
=
||a| |b| |c| sin θ cos φ|.
Thus
|a · (b × c)| is the volume of the parallelopiped. a · (b × c) is the volume or
the negative of the volume depending on whether {a,b,c} is a right or
left-handed system.
Note
that parentheses are unnecessary in a · b × c. There is only one way to
interpret the expression. If you did the dot product first then you would be
left with the cross product of a scalar and a vector which is meaningless. a · b
× c is called the scalar triple product.
Plane
Defined by Three Points. Three points which are
not collinear define a plane. Consider a plane that passes through the three
points a, b and c. One way of expressing that the point x lies in the plane is
that the vectors x − a, b − a and c − a are coplanar. (See Figure 2.11.) If the vectors are coplanar, then the
parallelopiped defined by these three vectors will have zero volume. We can
express this in an equation using the scalar triple product,
(x
− a) · (b − a) × (c − a) = 0.
Figure
2.11: Three points define a plane.
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