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Lines and Planes – Creating a Parallel Line (Example 1)
The equation of a line in general is given in general by:
\[r = {r_O} + tv\]
where \(r_O\) is the initial point, t is the time, and v is the vector direction.
Example Question: Find the line passing through (6,-5,2) and parallel to the vector \(\left\langle {1,3, – \frac{2}{3}} \right\rangle \)
So just use the point you were given, (6,-5,2) as \(r_O\), and use the vector provided, \(\left\langle {1,3, – \frac{2}{3}} \right\rangle \), as the direction vector.
Don’t forget to multiply the v direction vector by t.
\[r = \left\langle {6, – 5,2} \right\rangle + \left\langle {t,3t, – \frac{2}{3}t} \right\rangle \]
\[r = \left\langle {6 + t, – 5 + 3t,2 – \frac{2}{3}t} \right\rangle \]
You can think of each component as a dimension in 3-dimension space such as, x y and z for the first, second, and third components, respectively.
Try another example, where we make a line perpendicular to another line, in 3-dimension space!