QUESTION TWO (20 MARKS) (a) Show that the points \( A(2,-1,3), B(4,3,1) \) and \( C(3,1,2) \) are collinear. (b) Find the projection of the vectors \( \hat{i}+3 \hat{j}+7 \hat{k} \) on the vector \( 2 \hat{i}+6 \hat{j}+3 \hat{k} \). (c) If the directional derivative of \( \phi=a x^{2} y+b y^{2} z+c z^{2} x \) at the point \( (1,1,1) \) has maximum mag- nitude 15 in the direction parallel to the line find the values of \( a, b \) and \( c \). (d) Evaluate the area of the parallelogram whose two adjucent sides are determined by the vectors \( \hat{i}+2 \hat{j}+3 \hat{k} \) and \( 3 \hat{i}-2 \hat{j}+\hat{k} \). (e) Given the space curve \( x=3 \cos t, y=3 \sin t, z=4 t \). Compute the curvature \( k \) and the torsion \( \tau \). ( 4 mks .
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