Find the unit vector in the direction of the vector \(\vector{3, -4}, = 3\veci -4\vecj\text{.}\)
Section 2.1 Vectors in the Plane
References.
- OpenStax Calculus Volume 3, Section 2.1.
1
openstax.org/books/calculus-volume-3/pages/2-1-vectors-in-the-plane - Calculus, Early Transcendentals by Stewart, Section 12.2.
Introduction.
The earliest meaning of a vector relates to movement from one place to another (like a mosquito as a vector for malaria), and this leads to the geometrical idea of a vector describing the displacement from one location to another.
In this section we will consider the most basic case of vectors in the plane \(\mathbb{R}^2\text{,}\) where for example, points \(P_1(x_1,y_1)\) and \(P_2(x_2,y_2)\) determine the two dimensional vector going from point \(P_1\) to point \(P_2\text{,}\) denoted \(\veclong{P_1P_2}\text{.}\)
In algebraic terms, this is described by the changes in the values of each of the coordinates, so we denote this \(\veclong{P_1P_2}=\vector{x_2-x_1,y_2-y_1}\text{.}\) Note that only the change in position or displacement counts, so many different pairs of points represent the same vector. Thus we often name a vector without reference to a pair of endpoints, with an over-arrow like \(\vec{a}\text{,}\) or with boldface like \(\mathbf{a}\text{.}\)
Topics.
Subsection 2.1.1 Using Vectors as "Bundled Coordinates"
Any vector \(\vec{a} = \vector{x,y}\) can be considered with the origin \(O(0,0)\) as its starting point so that it ends at \(P(x,y)\text{:}\) \(\vec{a} = \veclong{OP}\text{.}\) Thus vectors give a convenient way to bundle the values of all the coordinates of a point \(P\) into a single object, and we can sometimes think of vectors as equivalent to points. But as we shall see, vectors have added features. Thus we call the set of all two dimensional vectors \(V_2\text{,}\) to distinguish from the set \(\mathbb{R}^2\) of points in the plane.
Subsection 2.1.2 The Length or Magnitude of a Vector
Vectors share some but not all of the properties of real numbers (which we now often call scalars to distinguish them from vectors) like length which resembles the absolute value of a scalar; addition; and and a form of multiplication. The length or magnitude of a vector \(\vec{a}=\vector{x,y}\) is the distance between its endpoints, denoted \(| \vec{a} |\) or \(\| \vec{a} \|\) to mimic the absolute value or magnitude of a scalar. Thus
\begin{equation}
\| \vec{a} \| = \| \vector{x,y} \| = \sqrt{x^2+y^2}\tag{2.1.1}
\end{equation}
Subsection 2.1.3 Adding Vectors
Since vectors describe displacements or movement from one point to another, one can combine several vectors, by making one move and then the other. If \(\vec{a} = \vector{a_1,a_2}\) and \(\vec{b} = \vector{b_1,b_2}\text{,}\) combining these two displacements changes the first [\(x\)] coordinate by \(a_1 + b_1\) and so on, so the combined displacement is described by the new vector \(\vector{a_1+b_1,a_2+b_2}\text{.}\) This is called the sum of the vectors \(\vec{a}\) and \(\vec{b}\text{,}\) and we write
\begin{equation}
\vec{a} + \vec{b} = \vector{a_1,a_2} + \vector{b_1,b_2} = \vector{a_1+b_1,a_2+b_2}.\tag{2.1.2}
\end{equation}
Clearly this is commutative like addition of real numbers:
\begin{equation}
\vec{a} + \vec{b} = \vec{b} + \vec{a}\tag{2.1.3}
\end{equation}
Also, there is a natural zero vector \(\vec{0} = \vector{0,0}\) representing no change in position, with
\begin{equation}
\vec{a} + \vec{0} = \vec{0} + \vec{a} = \vec{a}.\tag{2.1.4}
\end{equation}
Subsection 2.1.4 Scalar Multiples of Vectors
Repeated addition of copies of the same vector give natural number multiples of a vector, like
\begin{equation*}
2\vector{a_1,a_2} = \vector{a_1,a_2} + \vector{a_1,a_2} = \vector{2a_1,2a_2}.
\end{equation*}
This suggests the natural generalization to defining the number-by-vector product for any real number \(c\text{:}\)
\begin{equation}
c\vec{a} = c\vector{a_1,a_2} = \vector{c a_1,c a_2}\tag{2.1.5}
\end{equation}
Geometrically, \(c\vec{a}\) describes a displacement parallel to that described by \(\vec{a}\) but of magnitude scaled up or down by the factor \(|c|\text{,}\) and in the opposite direction if \(c\) is negative.
It is routine to check that this scalar-vector product is distributive in both factors and associative where it makes sense:
\begin{equation}
(c+d)\vec{a} = c\vec{a} + d\vec{a}, \quad c(\vec{a} + \vec{b}) = c\vec{a} + c\vec{b},
\quad (cd)\vec{a} = c(d\vec{a}).\tag{2.1.6}
\end{equation}
The Magnitude of a Product.
From the formula above for the length of a vector, it can be checked that the magnitude of a scalar-vector product is the product of the magnitudes:
\begin{equation}
\begin{split}
\|c\vec{a}\|
\amp= \sqrt{(ca_1)^2+(ca_2)^2}
\\
\amp= \sqrt{c^2(a_1^2+a_2^2)}
\\
\amp= \sqrt{c^2} \sqrt{a_1^2+a_2^2}
\\
\amp= |c| \|\vec{a}\|.
\end{split}\tag{2.1.7}
\end{equation}
Subsection 2.1.5 Vector Subtraction and Vector-Scalar Division
Subtraction as always is defined in terms of addition: \(\vec{a} - \vec{b}\) must be the vector that satisfies \((\vec{a} - \vec{b})+ \vec{b} = \vec{a}\text{,}\) and this has to be
\begin{equation}
\vec{a} - \vec{b} = \vector{a_1,a_2} - \vector{b_1,b_2} = \vector{a_1-b_1,a_2-b_2}.\tag{2.1.8}
\end{equation}
Likewise division by a scalar can be defined in terms of multiplication:
\begin{equation}
\vec{a}/c = \frac{1}{c}\vec{a}.\tag{2.1.9}
\end{equation}
Subsection 2.1.6 Basis Vectors
Two dimensional coordinates were described in terms of getting to a point \(P\) with a succession of two moves, parallel to each axis in turn. The displacement described by a vector can be broken up into two such displacements, which can in turn be written as multiples of displacements by distance one:
\begin{equation*}
\vec{a} = \vector{a_1,a_2} = \vector{a_1,0} + \vector{0,a_2}
\\
= a_1 \vector{1,0} + a_2 \vector{0,1}.
\end{equation*}
Thus any vector can be written in terms of the two special vectors appearing in the last line: \(\vec{a} = a_1\veci + a_2\vecj\) where
\begin{equation}
\veci = \vec{\imath} = \vector{1,0} \qquad \vecj = \vec{\jmath} = \vector{0,1}\tag{2.1.10}
\end{equation}
These are known as the standard basis vectors for the set \(V_2\) of vectors in the plane.
Subsection 2.1.7 Unit Vectors
Unit vectors are vectors of length one, like the standard basis vectors above. They are often used to indicate a direction of motion, without indicating the magnitude of that motion. For example, the above standard basis vectors indicate the directions "east" and "north". For any non-zero vector \(\vec{a}\text{,}\) there is a unique unit vector \(\vec{u}\) with the same direction,
\begin{equation*}
\vec{u} = \frac{1}{\|\vec{a}\|}\vec{a} = \frac{\vec{a}}{\|\vec{a}\|}
\end{equation*}
Exercise 2.1.1.
What is missing?
Although we have seen how to do with vectors much of what can be done with real numbers, a few things are missing: we have not defined
- a product of two vectors,
- a quotient of two vectors, or
- the inverse of a vector.
In subsequent sections we will see two versions of the product of vectors, but neither makes quotients or inverses possible.
Study Guide.
Study Section 2.1 of Calculus Volume 3; in particular:
7
openstax.org/books/calculus-volume-3/pages/2-1-vectors-in-the-plane- Theorem 2.1.
- Examples 2.1–2.8 and the Checkpoints following each.
- One or several exercises from each of the following ranges: 1–8, 11–14, 17–20.
