Line of intersection of two planes

A question is this type if and only if it asks to find the vector or Cartesian equation of the line where two planes intersect.

2 questions · Standard +0.6

4.04b Plane equations: cartesian and vector forms
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Edexcel FP3 Q6
11 marks Standard +0.8
  1. Referred to a fixed origin O , the points \(\mathrm { P } , \mathrm { Q }\) and R have coordinates \(( \mathbf { i } - 3 \mathbf { j } + \mathbf { k } ) , ( - 2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } )\) and \(( 3 \mathbf { j } - 5 \mathbf { k } )\) respectively. The plane \(\Pi _ { 1 }\) passes through \(\mathrm { P } , \mathrm { Q }\) and R . Find
    1. \(\overrightarrow { \mathrm { PQ } } \times \overrightarrow { \mathrm { QR } }\),
    2. a cartesian equation of \(\Pi _ { 1 }\).
    The plane \(\Pi _ { 2 }\) has equation \(\mathbf { r }\). ( \(\mathbf { i } + \mathbf { j } - \mathbf { k }\) ) \(= 6\). The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) intersect in the line I .
  2. Find a vector equation of I, giving your answer in the form ( \(\mathbf { r } - \mathbf { a }\) ) \(\times \mathbf { b } = \mathbf { 0 }\).
OCR FP3 Q6
9 marks Standard +0.3
The plane \(\Pi_1\) has equation \(\mathbf{r} = \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} + \mu \begin{pmatrix} -5 \\ -2 \end{pmatrix}\).
  1. Express the equation of \(\Pi_1\) in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [4] The plane \(\Pi_2\) has equation \(\mathbf{r} \cdot \begin{pmatrix} 7 \\ 1 \\ -3 \end{pmatrix} = 21\).
  2. Find an equation of the line of intersection of \(\Pi_1\) and \(\Pi_2\), giving your answer in the form \(\mathbf{r} = \mathbf{a} + t\mathbf{b}\). [5]