Work with Cartesian vectors

Extension: coefficient columns

A coordinate column need not describe a spatial point. In the source extension, a real polynomial $a_0+a_1x+\cdots+a_nx^n$ can be represented by the $(n+1)\times1$ coefficient column $\begin{pmatrix}a_0\\a_1\\\vdots\\a_n\end{pmatrix}$, using the fixed ordered basis $1,x,\ldots,x^n$. This represents polynomials of degree at most n, including zero leading coefficients.

Extend to three dimensions

Use three mutually perpendicular unit directions i,j,k along positive x,y,z. Then

$$\overrightarrow{OP}=x\mathbf i+y\mathbf j+z\mathbf k=\begin{pmatrix}x\\y\\z\end{pmatrix},\qquad\lvert\overrightarrow{OP}\rvert=\sqrt{x^2+y^2+z^2}.$$

The source example A(4,3,5) gives $\overrightarrow{OA}=(4,3,5)$ and $\lvert\overrightarrow{OA}\rvert=\sqrt{4^2+3^2+5^2}=5\sqrt2$.

Derive the three-dimensional length

To prove the formulas, the source marks A(x,0,0), B(0,y,0), C(0,0,z), R(x,y,0) and P(x,y,z). In the xy-plane, $\overrightarrow{OR}=x\mathbf i+y\mathbf j$. Since RP is parallel to the z-axis, $\overrightarrow{RP}=z\mathbf k$, so

$$\overrightarrow{OP}=\overrightarrow{OR}+\overrightarrow{RP}=(x\mathbf i+y\mathbf j)+z\mathbf k=x\mathbf i+y\mathbf j+z\mathbf k.$$

Two applications of Pythagoras give

$$\lvert\overrightarrow{OR}\rvert=\sqrt{x^2+y^2},\qquad\lvert\overrightarrow{OP}\rvert=\sqrt{\lvert\overrightarrow{OR}\rvert^2+\lvert\overrightarrow{RP}\rvert^2}=\sqrt{(\sqrt{x^2+y^2})^2+z^2}=\sqrt{x^2+y^2+z^2}.$$

The orthonormal basis columns are $\mathbf i=\begin{pmatrix}1\\0\\0\end{pmatrix}$, $\mathbf j=\begin{pmatrix}0\\1\\0\end{pmatrix}$, $\mathbf k=\begin{pmatrix}0\\0\\1\end{pmatrix}$. For a right-handed system, curl the right-hand fingers from +x towards +y; the thumb points along +z. Viewed from +z towards O, that quarter-turn is anticlockwise. The projected drawing does not show true on-screen lengths or right angles.

Keep the six coordinate results together

For $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$:

1. Position vector.

$$\overrightarrow{OA}=x_1\mathbf i+y_1\mathbf j+z_1\mathbf k=\begin{pmatrix}x_1\\y_1\\z_1\end{pmatrix}$$

2. Magnitude.

$$\lvert\overrightarrow{OA}\rvert=\sqrt{x_1^2+y_1^2+z_1^2}$$

3. Unit direction, when A≠O.

$$\dfrac{\overrightarrow{OA}}{\lvert\overrightarrow{OA}\rvert}=\dfrac1{\sqrt{x_1^2+y_1^2+z_1^2}}\begin{pmatrix}x_1\\y_1\\z_1\end{pmatrix}$$

4. Scalar multiplication.

$$k\overrightarrow{OA}=\begin{pmatrix}kx_1\\ky_1\\kz_1\end{pmatrix}$$

5. Equality of vectors.

$$\overrightarrow{OA}=\overrightarrow{OB}\iff x_1=x_2,\ y_1=y_2,\ z_1=z_2$$

All three components must agree. 6. Addition and subtraction.

$$\overrightarrow{OA}\pm\overrightarrow{OB}=\begin{pmatrix}x_1\\y_1\\z_1\end{pmatrix}\pm\begin{pmatrix}x_2\\y_2\\z_2\end{pmatrix}=\begin{pmatrix}x_1\pm x_2\\y_1\pm y_2\\z_1\pm z_2\end{pmatrix}$$