Multivariate Calculus: multivariate calculus – #27255


. (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]

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