8.04e Scalar triple product: volumes of tetrahedra and parallelepipeds

3 questions

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Edexcel FP1 2019 June Q7
10 marks Challenging +1.2
  1. With respect to a fixed origin \(O\), the points \(A\), \(B\) and \(C\) have coordinates \(( 3,4,5 ) , ( 10 , - 1,5 )\) and ( \(4,7 , - 9\) ) respectively.
The plane \(\Pi\) has equation \(4 x - 8 y + z = 2\) The line segment \(A B\) meets the plane \(\Pi\) at the point \(P\) and the line segment \(B C\) meets the plane \(\Pi\) at the point \(Q\).
  1. Show that, to 3 significant figures, the area of quadrilateral \(A P Q C\) is 38.5 The point \(D\) has coordinates \(( k , 4 , - 1 )\), where \(k\) is a constant.
    Given that the vectors \(\overrightarrow { A B } , \overrightarrow { A C }\) and \(\overrightarrow { A D }\) form three edges of a parallelepiped of volume 226
  2. find the possible values of the constant \(k\).
Edexcel FP1 2020 June Q3
9 marks Standard +0.3
  1. The points \(A , B\) and \(C\), with position vectors \(\mathbf { a } = 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } , \mathbf { b } = \mathbf { i } + 4 \mathbf { j } + 5 \mathbf { k }\) and \(\mathbf { c } = - 2 \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k }\) respectively, lie on the plane \(\Pi\)
    1. Find \(\overrightarrow { A B } \times \overrightarrow { A C }\)
    2. Find an equation for \(\Pi\) in the form r.n \(= p\)
    The point \(D\) has position vector \(8 \mathbf { i } + 7 \mathbf { j } + 5 \mathbf { k }\)
  2. Determine the volume of the tetrahedron \(A B C D\)
OCR Further Additional Pure 2018 March Q2
5 marks Standard +0.8
2 Four points \(A , B , C\) and \(D\) have coordinates \(( 1,2,5 ) , ( 3,4 , - 4 ) , ( 6,2,3 )\) and \(( 0,3,7 )\) respectively. Find the volume of tetrahedron \(A B C D\).