13.1: Vectors

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Franz S. Hover & Michael S. Triantafyllou
Massachusetts Institute of Technology via MIT OpenCourseWare

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Definition

We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in three-space, we write a vector in terms of its components with respect to a reference system as

\[ \vec{a} \, = \, \begin{Bmatrix} \,2\, \\[4pt] \,1\, \\[4pt] \,7\, \end{Bmatrix}. \]

The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.

Vector addition: \begin{align} \vec{a} + \vec{b} \, &= \, \vec{c}\\[4pt][5 pt] \begin{Bmatrix} \,2\, \\[4pt] \,1\, \\[4pt] \,7\, \end{Bmatrix} + \begin{Bmatrix} \,3\, \\[4pt] \,3\, \\[4pt] \,2\, \end{Bmatrix} \, &= \, \begin{Bmatrix} \,5\, \\[4pt] \,4\, \\[4pt] \,9\, \end{Bmatrix} \end{align} Graphically, addition is stringing the vectors together head to tail.
Scalar multiplication: \[ -2 \times \begin{Bmatrix} \,2\, \\[4pt] \,1\, \\[4pt] \,7\, \end{Bmatrix} \, = \, \begin{Bmatrix} \,-4\, \\[4pt] \,-2\, \\[4pt] -14 \end{Bmatrix} \]

\(\PageIndex{2}\): Vector Magnitude

The total length of a vector of dimension \(m\), its Euclidean norm, is given by

\[ || \vec{x} || \, = \, \sqrt{\sum_{i=1}^{m} x_i^2}. \] This scalar is commonly used to normalize a vector to length one.

\(\PageIndex{3}\): Vector Dot or Inner Product

The dot product of two vectors is a scalar equal to the sum of the products of the corresponding components: \[\vec{x} \cdot \vec{y} \, = \, \vec{x}^T \vec{y} \, = \, \sum_{i=1}^{m} x_i y_i.\] The dot product also satisfies \[\vec{x} \cdot \vec{y} \, = \, ||\vec{x}|| ||\vec{y}|| \cos \theta, \] where \(\theta\) is the angle between the vectors.

\(\PageIndex{4}\): Vector Cross Product

The cross product of two three-dimensional vectors \(\vec{x}\) and \(\vec{y}\) is another vector \(\vec{z}\), written as \(\vec{x} \times \vec{y} = \vec{z}\). Vector \(\vec{z}\)'s

direction is normal to the plane formed by the other two vectors,
direction is given by the right-hand rule, rotating from \(\vec{x}\) to \(\vec{y}\),
magnitude is the area of the parallelogram formed by the two vectors – the cross product of two parallel vectors is zero – and
(signed) magnitude is equal to \(||\vec{x}|| ||\vec{y}|| \sin \theta\), where \(\theta\) is the angle between the two vectors, measured from \(\vec{x}\) to \(\vec{y}\).

In terms of their components,

\[ \vec{x} \times \vec{y} \, = \, \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\[4pt] x_1 & x_2 & x_3 \\[4pt] y_1 & y_2 & y_3 \end{vmatrix} \, = \, \begin{Bmatrix} (x_2 y_3 - x_3 y_2) \hat{i} \\[4pt] (x_3 y_1 - x_1 y_3) \hat{j} \\[4pt] (x_1 y_2 - x_2 y_1) \hat{k} \end{Bmatrix}. \]