Find the vector and the Cartesian equations of the line through the point (5, 2, – 4) and which is parallel to the vector \(3 \hat{i}+2 \hat{j}-8 \hat{k}\)

4

(a)

Show that the equation of a plane which meets the axes in A,B and C and the given centroid of triangle ABC is the point \((\alpha ,\beta ,\gamma )\) is \(\frac { x }{ \alpha } +\frac { y }{ \beta } +\frac { z }{ \gamma } =3\)

4

(a)

If a variable line in two adjacent positions has direction cosines l,m,n,\(l+\delta l,m+\delta m,n+\delta n\) and show that the small angle \(\delta \theta \) between two positions is given by \({ (\delta \theta ) }^{ 2 }={ (\delta l) }^{ 2 }+(\delta m)^{ 2 }+{ (\delta n) }^{ 2 }\). 

4

(a)

Show that the lines \(\frac { x-1 }{ 2 } =\frac { y-2 }{ 3 } =\frac { z-3 }{ 4 } \) and \(\frac { x-4 }{ 5 } =\frac { y-1 }{ 2 } =z\) intersect Also, find the point of intersection.

4

(a)

Find the equation of the plane which is perpendicular to the plane 5x + 3y + 6z + 8 = 0 and which contains the line of intersection of the planes  x + 2y + 3z - 4 = 0 and 2x +y - z + 5 = 0.

4

(a)