Question 4 15 marks Paper 1 Section A Hard No calculator Free
Maximum mark: 15
The vertices of a triangle ABC have coordinates given by \( A(-1, 2, 3) \), \( B(4, 1, 1) \) and \( C(3, -2, 2) \).
(a) (i) Find the lengths of the sides of the triangle.[3]
(ii) Find \( \cos B\hat{A}C \).[2]
(b) (i) Show that \( \overrightarrow{BC} \times \overrightarrow{CA} = -7\mathbf{i} - 3\mathbf{j} - 16\mathbf{k} \).[2]
(ii) Hence, show that the area of the triangle ABC is \( \dfrac{1}{2}\sqrt{314} \).[1]
(c) Find the Cartesian equation of the plane containing the triangle ABC.[2]
(d) Find a vector equation of (AB).[1]
The point D on (AB) is such that \( \overrightarrow{OD} \) is perpendicular to \( \overrightarrow{BC} \), where O is the origin.
(e) (i) Find the coordinates of D.[3]
(ii) Show that D does not lie between A and B.[1]
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