Vectors

It's a mathematical term, represented by an arrow with both direction and magnitude.

The Essentials

Vectors can be defined by their components:

\[ \vec{v}\hspace{12pt}=\hspace{12pt} \langle1,2,2\rangle\hspace{12pt}=\hspace{12pt}\begin{bmatrix} 1\\2\\2 \end{bmatrix}\hspace{12pt}=\hspace{12pt}1\hat{i}+2\hat{j}+2\hat{k} \]

They can also be defined by their magnitude and direction, whether in two dimensions or any other number of dimensions:

\[ r=5, \theta=63^\circ\hspace{24pt}r=5,\theta=63,\phi=68 \]

When vectors are added (and subtracted) the first components are added, the second components, etc.:

\[ \begin{bmatrix} a\\b\\c \end{bmatrix}+\begin{bmatrix} d\\e\\f \end{bmatrix}=\begin{bmatrix} a+d\\b+e\\c+f \end{bmatrix} \]

When visualizing added vectors, one vector is placed where the other ends, and the end of the second is where the end of the added vector goes:

\[ \textcolor{red}{\langle 1,2\rangle} + \textcolor{blue}{\langle2,-1\rangle} =\textcolor{ForestGreen}{\langle3,1\rangle} \]
calculus 3 vectores figure 1

The magnitude of a vector is found using this formula:

\[ ||\vec{v}||=\sqrt{v_1^2+v_2^2+\ldots} \]

If the vector is two dimensional, the angle can be found using this equation:

\[ \theta=\arctan\left(\frac{v_2}{v_1}\right) \]

A unit vector is a vector that has a magnitude of one. If a vector is divided by its magnitude, it becomes a unit vector in the same direction as the original vector.

The distance between two vectors, \( \vec{u} \) and \( \vec{v} \), is given by the formula:

\[ D=\sqrt{(v_1-u_1)^2+(v_2-u_2)^2+\ldots} \]

The Dot Product

The dot product of two vectors is accomplished by adding the products of the corresponding components of the two vectors:

\[ \vec{u}\cdot\vec{v}=u_1v_1 + u_2v_2 + \ldots \]

Note that the result of a dot product is a scalar. The dot product can also be found by multiplying the magnitude of the vectors by the cosine of the angle between them:

\[ \vec{u}\cdot\vec{v}=||\vec{u}||||\vec{v}||cos(\theta) \]

Two vectors are considered orthogonal to each other if their dot product is zero.

The Cross Product

The cross product can only be performed on two vectors that are three (or seven) dimensional. The result is a vector that is orthogonal to both of the original vectors, whether those vectors were orthogonal to each other or not. Unlike the dot product, order matters for the cross product. The cross product is given by this formula:

\[ \vec{u}\times\vec{v}=\langle u_2v_3-u_3v_2, -(u_1v_3-u_3v_1), u_1v_2-u_2v_1\rangle \]

If one knows a little bit about matrices, the formula for the cross product can be remembered by taking the determinant of a three by three matrix whose row vectors are, in order, the arbitrary vector, then the first vector, then the second vector:

\[ \vec{u}\times\vec{v}=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{array}\right| \]

The magnitude of the cross product can be found using this formula:

\[ ||\vec{u}\times\vec{v}||=||\vec{u}||||\vec{v}||sin(\theta) \]

Practice

  1. If \( \vec{u}=\langle e^\pi, \cos(5), \sqrt{67} \rangle \) and \( \vec{v}=\langle\sqrt[6]{7}, \pi, e+e^3 \rangle \), find \( \vec{u}\times\vec{v}\cdot\vec{u} \). (Hint: If you are doing math, then you are doing it wrong)
  2. Find the dot product: \( \langle1,2,34,5\rangle\cdot\langle5,43,2,1\rangle \)
  3. Find the cross product: \( \langle1,2,3\rangle\cdot\langle-1,3,-2\rangle \)

Solutions:

  1. 0
  2. 164
  3. \( \langle-13,-1,5\rangle \)

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