# Multivariate Calculus: multivariate calculus – #27255

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]