In this section, I introduce/review the basic arithmetic of vectors. I’ll define the operations in \(\RR^3\text{;}\) for \(\RR^2\text{,}\) all the operations are the same just without the third component. The exception is the cross-product, which is unique to \(\RR^3\text{.}\)
Definition1.2.1.
The sum of two vectors \(u\) and \(v\) in \(\RR^3\) is the sum taken component wise.
\begin{equation*}
u + v = (u_1,u_2,u_3) + (v_1,v_2,v_3) = (u_1 + v_1, u_2 +
v_2, u_3 + v_3)
\end{equation*}
The sum is visualized by placing the start of the second vector at the end of the first, as in Figure 1.2.2. Note that I can only add two vectors in the same dimension. I can’t add a vector in \(\RR^2\) to a vector in \(\RR^3\text{.}\)
Figure1.2.2.Visualizing Vector Addition
Definition1.2.3.
If \(u\) is a vector in \(\RR^3\) and \(a \in \RR\) is a real number, then the scalar multiplication of \(u\) and \(a\) is multiplication by \(a\) in each component of \(u\text{.}\) By convention, scalar multiplication is written with the scalar on the left of the vector.
\begin{equation*}
au = a (u_1, u_2, u_3) = (au_1, au_2, au_3)
\end{equation*}
Though there will be other ‘multiplications’ to come, it is mostly true that I can’t multiply vectors together in any way reminiscent of numbers. Instead, I can only multiply by scalars. Scalar multiplication is visualized by scaling the vector by the value of the scalar. (Hence the term ‘scalar’!) If the scalar is negative, the direction is also reversed, as in Figure 1.2.4.
Figure1.2.4.Visualizing Scalar Multiplication
Scalar multiplication also lets me define the difference between vectors.
Definition1.2.5.
The difference between two vectors \(u\) and \(v\) is the vector \(u + (-1)v\text{,}\) defined using addition and scalar multiplication. This works out to be componentwise subtraction.
\begin{equation*}
u - v = u + (-1) v= (u_1, u_2,u_3) + (-1) (v_1, v_2, v_3)
= (u_1 - v_1, u_2 - v_2, u_3 - v_3)
\end{equation*}
Definition1.2.6.
The length of a vector \(u\) in \(\RR^3\) is written \(|u|\) and is given by a generalized form of the Pythagorean rule for right triangles.
This length is also called the norm of the vector. A vector of length one is called a unit vector.
If I think of vectors as directions from the origin towards a point, this definition of length gives exactly what I expect: the physical length of that arrow in \(\RR^2\) and \(\RR^3\text{.}\) Note also that \(|u| = 0\) only if \(u\) is the zero vector. All other vectors have a positive length.
Often the square root is annoying, and I find it convenient to work with the square of the length.
The notions of length and difference allow me to define the distance between two vectors.
Definition1.2.7.
The distance between two vectors \(u\) and \(v\) in \(\RR^n\) is the length of their difference: \(|u-v|\text{.}\)
You can check from the definition that \(|u-v| = |v-u|\text{,}\) so distance doesn’t depend on which comes first. If \(|\cdot|\) were absolute value in \(\RR\text{,}\) this definition would match the notion of distance between numbers on the number line. Difference and length are visualized in Figure 1.2.8.
Figure1.2.8.Visualizing Distance Between Vectors
Subsection1.2.2The Dot Product
Earlier, I said that I couldn’t multiply two vectors together. That’s mostly true: there is no general product of two vectors \(uv\text{,}\) which is still a vector, as least none that has any useful or reasonable geometric meaning. However, there are other kinds of ‘multiplication’ which combine two vectors. The operation defined here starts with two vectors, but the result is a scalar.
Definition1.2.9.
The dot product or inner product or scalar product of two vectors \(u\) and \(v\) is given by the following formula.
\begin{equation*}
u \cdot v = (u_1,u_2,u_3) \cdot (v_1,v_2,v_3) =
u_1 v_1 + u_2 v_2 + u_3 v_3
\end{equation*}
The dot product relates to angles between vectors in an important way.
Proposition1.2.10.
If \(u\) and \(v\) are two vectors in \(\RR^n\text{,}\) then the angle between them \(\theta\) satisfies this equation.
The output of the inverse cosine is an angle in the range \([0,
\pi) \text{.}\) This convention works well for vectors, since the angle between them can always be chosen to be in this range. A useful implication of this relationship between angles and dot-products is that dot products can be used to identify perpendicular vectors.
Definition1.2.11.
Two vectors \(u\) and \(v\) in \(\RR^3\) are orthogonal or perpendicular or normal to each other if and only if \(u \cdot
v = 0\text{.}\)
It will be useful to know how the dot product interacts with the other previously defined operations. Here are some properties.
Proposition1.2.12.
Let \(u,v,w\) be vectors in \(\RR^3\) and let \(a\) be a scalar in \(\RR\text{.}\) The following four identities are true for all vectors and scalars.
\begin{align*}
u \cdot v \amp = v \cdot u \amp \text{Commutative} \\
u \cdot u \amp = |u|^2 \amp \\
u \cdot (v+w) \amp = u \cdot v + u \cdot w \amp
\text{Distributive} \\
u \cdot (av) \amp = (au) \cdot v = a (u \cdot v)
\end{align*}
Subsection1.2.3The Cross Product
The cross product is a unique operations to \(\RR^3\text{.}\)
Definition1.2.13.
Let \(u = (u_1,u_2,u_3)\) and \(v = (v_1,v_2,v_3)\) be two vectors in \(\RR^3\text{.}\) The cross product of \(u\) and \(v\) is written \(u \times v\) and defined by the following formula.
\begin{equation*}
u \times v = (u_2v_3 - u_3v_2, u_3v_1 - u_1v_3, u_1v_2 -
u_2v_1)
\end{equation*}
The cross product differs from the dot product in several important ways. First, it produces a new vector in \(\RR^3\text{,}\) not a scalar. For this reason, when working in \(\RR^3\text{,}\) the dot product is often referred to as the scalar product and the cross product as the vector product. Second, the dot product measures, in some sense, the similarity of two vectors. The cross product measures, in some sense, the difference between two vectors. The cross product has a greater magnitude if the vectors are closer to being perpendicular. The dot product is zero if two vectors are perpendicular, but the cross product is zero if two vectors are parallel.
I could do a general calculation to show that \(v \cdot (u
\times v) = 0\text{.}\) Since a dot product of two vectors is zero if and only if the vectors are perpendicular, the vector \(v
\times u\) is perpendicular to both \(u\) and \(v\text{.}\) This turns out to be a very useful property of the cross product: it produces a result that is perpendicular to both original vectors. This is part of the reason that is is unique to \(\RR^3\text{:}\) in three dimensions, if you have two set (non-parallel) direction, there is a unique direction that is perpendicular to both. The cross product constructs that unique direction.
Finally, a calculation from the definition shows that \(u
\times v = -(v \times u)\text{.}\) So far, multiplication of scalars and the dot product of vectors have not depended on order. The cross product is one of many products in mathematics which depends on order. If I change the order of the cross product, I introduce a negative sign. An operation which changes sign when the order changes is called anti-commutative.
Subsection1.2.4Angular Motion
An important application of the cross product is found in describing rotational motion. Linear mechanics describes the motion of an object through space, but rotational mechanics describes the rotation of an object independent of its movement through space. A force on an object can cause both kinds of movement, obviously. The following table summarizes the parallel questions of linear motion and rotational motion in \(\RR^3\text{.}\)
Linear Motion
Rotational Motion
Moving in a straight line
Continual spinning
Direction of motion
Axis of spin
Force
Torque
Momentum
Angular Momentum
Mass (resistance to motion)
Moment of Inertia (resistance to spin)
Velocity
Frequency (Angular Velocity)
Acceleration
Angular Acceleration
How should I describe torque? If there is a linear force applied to an object which can only rotate around an axis, and if the linear force is applied at a distance \(r\) from the axis, I can think of the force \(F\) and the distance \(r\) as vectors. The torque is then \(\tau = r \times F\text{.}\) Since the cross product is zero when vectors are parallel and large when vectors are perpendicualr, a force perpendicular to the radius gives the greatest angular acceleration. That makes sense. If \(F\) and \(r\) share a direction, then the force is pushing directly along the axis and no rotation can occur.
The use of cross products in rotational dynamics is ubiquitous. In fluid dynamics, local rotational motion of the fluid result in turbulence, vortices and similar effects. Tornadoes and hurricanes are particularly extreme examples of vortices. All the descriptions of the force and motion of these vortices involve cross products in the vectors describing the fluid. In this course, I will specifically use cross products to discuss angular accelration for parametric curves in Subsection 2.2.8.
