Contents
<aside> <img src="/icons/book_orange.svg" alt="/icons/book_orange.svg" width="40px" /> A compendium for Mathematics NS1 written by Martin Deraas has been published on the Amazon store. It is available for purchase as a paperback for £9.95 (120 NOK), and as a PDF document on Google Drive/Adobe Acrobat for NOK 100.
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<aside> <img src="https://image.flaticon.com/icons/svg/1828/1828885.svg" alt="https://image.flaticon.com/icons/svg/1828/1828885.svg" width="40px" /> A unit vector is often denoted by a lowercase letter with a circumflex, or "hat".
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A vector is something that has both a magnitude and direction (unlike scalars which only have a magnitude).
It has two or more dimensions.
$$ \vec v=\langle x_1,y_1,z_1\rangle $$
The vector $v = \langle x,y,z\rangle$ has the length (scalar):
$$ \vert\vec v\vert=\sqrt{x^2+y^2+z^2} $$
Zero (or null) vector
$$ \vert\vec 0\vert=0 $$
Unit (or direction) vector: A vector in a normed vector space whose norm is 1, or a Euclidean vector of length one.
$$ \vert\vec e\vert=1 $$
û is the unit vector in the direction of a non-zero vector u;
$$ \hat u={\vec u\over\vert\vec u\vert} $$
The standard basis, also called natural basis, for a Euclidean vector space equipped with a Cartesian coordinate system is the set of vectors whose coordinates are all 0, except one that equals 1.
$$ \bf\hat i=\langle 1,0,0\rangle=\begin{bmatrix}1\\0\\0\end{bmatrix} $$
$$ \bf\hat j=\langle 0,1,0\rangle=\begin{bmatrix}0\\1\\0\end{bmatrix} $$
$$ \bf\hat k=\langle 0,0,1\rangle=\begin{bmatrix}0\\0\\1\end{bmatrix} $$
In addition to being the as a sum of three vectors that are not parallel, any arbitrary vector can be expressed as the sum of its coordinates multiplied with their respective unit vector;
$$ \begin{alignedat}{4} &\vec v=&x\vec a&+y\vec b&+z\vec c\\ \vec v=[x,y,z]\Leftrightarrow\space &\vec v=&x\hat i&+y\hat j&+z\hat k \end{alignedat} $$
Addition and subtraction
$$ \begin{aligned} [x_1,y_1]+[x_2,y_2] & =[x_1+x_2,y_1+y_2]\\ [x_1,y_1]-[x_2,y_2] & =[x_1-x_2,y_1-y_2] \end{aligned} $$
Multiplication
$$ t[x,y]=[tx,ty] $$
Dot (or scalar) product
$$ [x_1,y_1,z_1]\cdot[x_2,y_2,z_2]=x_1x_2+y_1y_2+z_1z_2 $$
$$ \vec a\cdot\vec b=\vert\vec a\vert\cdot\vert\vec b\vert\cdot\cos\space\theta $$
From this we can conclude that:
$$ \vec a\perp\vec b\Leftrightarrow \vec a\cdot\vec b=0 $$
$$ \vec a^2=\vert\vec a\vert^2 $$
Cross product
$$ [x_1,y_1,z_1]\cdot[x_2,y_2,z_2]=\langle y_1z_2-z_1y_2,-(x_1z_2-z_1x_2),x_1y_2-y_1x_2\rangle $$
Magnitude of the cross product
$$ \vec a\cdot\vec b=\vert\vec a\vert\cdot\vert\vec b\vert\cdot\sin\space\theta $$
$$ \vert\vec v+\vec w\vert\space{\footnotesize\le}\space\vert v\vert+\vert w\vert $$
$$ \vert v\vert=\sqrt{x^2+y^2+z^2} $$
$$ \vert v+w\vert=\sqrt{(x_2+x_1)^2+(y_2+y_1)^2+(z_2+z_1)^2} $$
$$ \vert v-w\vert=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2} $$
Circle equation
For a circle of radius r with centre (initial point) $(a,b,c)$
$$ (x-a)^2 + (y-b)^2+(z-c)^2=r^2 $$
The points A, B, and C of a vector space belong to the same line if the vector AC is parallel to the vector AB, so that:
$$ \overrightarrow{AC}=t\overrightarrow{AB} $$
The parametric (or vector) equation of a line l that intersects the point (x_0,y_0,z_0) and is parallel to the vector v = ⟨a,b,c⟩, is:
$$ l:\begin{cases}\begin{alignedat}{3}x&=x_0&+&at\\y&=y_0&+&bt\\z&=z_0&+&ct\end{alignedat}\end{cases} $$
If a, b, and c are nonzero numbers, then the symmetric equation of the line are:
$$ {x-x_0\over a}={y-y_0\over b}={z-z_0\over c} $$
The addition of two vectors a and b. Photo: Benjamin D. Esham (bdesham), Public Domain, Wikimedia.
The subtraction of two vectors a and b. Photo: Benjamin D. Esham (bdesham), Public Domain, Wikimedia.