(a) The point (2, 1, 0) is rotated through 180° abou

Question:

.	(a)	The point \[\left( 2,\,\,1,\,\,0 \right)\] is rotated through 180° about the axis parallel to \[(\underset{\raise0.3em\hbox{\]\smash{\scriptscriptstyle-}$}}{i}+\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{k})$, which passes through the origin.
		(i)	Use quaternions to calculate the image point of the rotation.
		(ii)	If the image point undergoes a further rotation of 90⁰ about an axis parallel to \[\underset{\raise0.3em\hbox{\]\smash{\scriptscriptstyle-}$}}{j}$, again passing through the origin, determine the single equivalent rotation axis and angle of rotation for the double rotation.	[11]
	(b)	The points \[P(7,\,8,\,13)\] and \[Q(-1,\,6,\,3)\] are projected orthogonally onto the plane \[x+2y+3z=6\].
		(i)	Obtain the coordinates of the image points \[{P}'\] and \[{Q}'\].
		(ii)	Show that the length of the line segment \[{P}'{Q}'\] is half of the length of the line segment \[PQ\].	[9]