Skip to main content
\(\require{cancel}\)
Physics LibreTexts

13.2: Triangles

  • Page ID
    6869
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    I shall start with a geometric theorem involving triangles, which will be useful as we progress towards our aim of computing orbital elements.


    Figure 13.1.png
    \(\text{FIGURE XIII.1}\)

    Figure \(\text{XIII.1}\) shows three coplanar vectors. It is clearly possible to express \(\textbf{r}_2\) as a linear combination of the other two. That is to say, it should be possible to find coefficients such that

    \[\textbf{r}_2 = a_1 \textbf{r}_1 + a_3 \textbf{r}_3 . \label{13.2.1}\]

    The notation I am going to use is as follows:

    • The area of the triangle formed by joining the tips of \(\textbf{r}_2\) and \(\textbf{r}_3\) is \(A_1\).
    • The area of the triangle formed by joining the tips of \(\textbf{r}_3\) and \(\textbf{r}_1\) is \(A_2\).
    • The area of the triangle formed by joining the tips of \(\textbf{r}_1\) and \(\textbf{r}_2\) is \(A_3\).

    To find the coefficients in Equation \ref{13.2.1}, multiply both sides by \(\textbf{r}_1 \times\):

    \[\textbf{r}_1 \times \textbf{r}_2 = a_3 \textbf{r}_1 \times \textbf{r}_3 . \label{13.2.2}\]

    The two vector products are parallel vectors (they are each perpendicular to the plane of the paper), of magnitudes \(2A_3\) and \(2A_2\) respectively. (\(2A_3\) is the area of the parallelogram of which the vectors \(\textbf{r}_1\) and \(\textbf{r}_2\) form two sides.)

    \[\therefore a_3 = A_3/A_2 . \label{13.2.3}\]

    Similarly by multiplying both sides of Equation \ref{13.2.1} by \(\textbf{r}_3 \times\) it will be found that

    \[a_1 = A_1/ A_2 . \label{13.2.4}\]

    Hence we find that

    \[A_2 \textbf{r}_2 = A_1 \textbf{r}_1 + A_3 \textbf{r}_3 . \label{13.2.5}\]


    13.2: Triangles is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.