# Multivariate Calculus: multivariate calculus – #27256

Question:

 (a) The vertices of a parallelogram plane segment are $A\left( 3,\,4,\,1 \right)$, $B\left( 0,\,1,\,4 \right)$, $C\left( 2,\,0,\,3 \right)$and$D\left( 5,\,3,\,0 \right)$. A ray starts from $\left( 0,\,0,\,1 \right)$ and ends at $\left( 4,\,4,\,5 \right)$. (i) Taking the non-parallel sides as AB and AD, determine the parametric vector equation of the parallelogram plane segment. Determine also the parametric vector equation of the ray. (ii) Hence obtain the coordinates of the point of intersection of the ray with the parallelogram plane segment. (iii) State, with a reason, whether or not the ray would intersect the triangular segment ADC. [10] (b) A viewing window is a rectangle with vertices $A(1,1)$, $B\left( 9,-5 \right)$, $C\left( 12,-1 \right)$ and $D\left( 4,5 \right)$. (i) Obtain a concatenation of matrices which will map this window onto the normalised device screen in such a way that aspect ratio is preserved. Simplify your product to a single matrix. (ii) What addition matrix would have to be appended to the concatenation you wrote down in part (i) if it were required that the image should be centred vertically in the normalised device screen? (Do not carry out any further simplification of the matrix product) [10]