7 Fig. 7 shows a tetrahedron ABCD . The coordinates of the vertices, with respect to axes \(\mathrm { O } x y z\), are \(\mathrm { A } ( - 3,0,0 ) , \mathrm { B } ( 2,0 , - 2 ) , \mathrm { C } ( 0,4,0 )\) and \(\mathrm { D } ( 0,4,5 )\).
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\caption{Fig. 7}
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- Find the lengths of the edges AB and AC , and the size of the angle CAB . Hence calculate the area of triangle ABC .
- (A) Verify that \(4 \mathbf { i } - 3 \mathbf { j } + 10 \mathbf { k }\) is normal to the plane ABC .
(B) Hence find the equation of this plane. - Write down a vector equation for the line through D perpendicular to the plane ABC . Hence find the point of intersection of this line with the plane ABC .
The volume of a tetrahedron is \(\frac { 1 } { 3 } \times\) area of base × height.
- Find the volume of the tetrahedron ABCD .