Three-dimensional position and displacement vectors — 9709 P3
Apply component operations and magnitude consistently in 3D.
Notes and key methods
A position vector records three coordinates from a common origin. Subtract endpoint position vectors to obtain a displacement and add displacements head-to-tail.
For \(\displaystyle v=(a,b,c), |v|=\sqrt{a^{2}+b^{2}+c^{2}}. A\) nonzero v can be normalised to v/|v|. A midpoint averages all three coordinates, not just those visible in a 2D sketch.
Check the conditions
All positions use one origin.
The displacement used for a unit direction is nonzero.
Three worked examples
Example 1 · Position and displacement vectors in three dimensions
A has position vector \(\left[\begin{matrix}2\\2\\2\end{matrix}\right]\) and B has position vector \(\left[\begin{matrix}4\\5\\6\end{matrix}\right]\).
Find AB, its length and a unit vector in its direction.
Find the z-component of AB.
Find \(\displaystyle |AB|^{2}\).
Find the midpoint x-coordinate.
Show the reasoning
(a)
Use b−a, its positive magnitude, and then normalise.
Example 3 · Position and displacement vectors in three dimensions
A has position vector \(\left[\begin{matrix}4\\2\\4\end{matrix}\right]\) and B has position vector \(\left[\begin{matrix}5\\7\\10\end{matrix}\right]\).
Find AB, its length and a unit vector in its direction.
Find the z-component of AB.
Find \(\displaystyle |AB|^{2}\).
Find the midpoint x-coordinate.
Show the reasoning
(a)
Use b−a, its positive magnitude, and then normalise.