Components of Vector

To describe a vector's direction precisely, it is written as a linear combination of three special basis vectors. Taking $A(1,0,0),B(0,1,0),C(0,0,1)$ on the X, Y, Z axes, $|\overrightarrow{OA}| =|\overrightarrow{OB}|=|\overrightarrow{OC}|=1$; these three unit vectors along the axes are named $\hat \imath,\hat\jmath,\hat k$ (also called the standard unit vectors). Any vector along the X-axis is a

scalar multiple of $\hat\imath$ (e.g. $3\hat\imath$ is a vector of magnitude 3 along $OX$); likewise for

$\hat\jmath,\hat k$ along $OY,OZ$.

Theorem (unique decomposition along three non-coplanar vectors). If $\bar a,\bar b,\bar c$ are

non-coplanar, any vector $\bar r$ in space can be written uniquely as $\bar r=x\bar a+y\bar b+z\bar c$.

Sketch: build a parallelepiped through the common initial point using lines parallel to the planes of pairs

of the three vectors; the triangle law applied through the parallelepiped's vertices gives

$\bar r=x\bar a+y\bar b+z\bar c$ for some scalars $x,y,z$; uniqueness again follows by subtracting two …