4.04b Plane equations: cartesian and vector forms

277 questions

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Pre-U Pre-U 9795/1 2015 June Q10
11 marks Standard +0.8
  1. Find a vector equation for the line of intersection of the planes with cartesian equations $$x + 7y - 6z = -10 \quad \text{and} \quad 3x - 5y + 8z = 48.$$ [5]
  2. Determine the value of \(k\) for which the system of equations \begin{align} x + 7y - 6z &= -10
    3x - 5y + 8z &= 48
    kx + 2y + 3z &= 16 \end{align} does not have a unique solution and show that, for this value of \(k\), the system of equations is inconsistent. [6]
Pre-U Pre-U 9795/1 2018 June Q13
18 marks Challenging +1.2
The planes \(\Pi_1\) and \(\Pi_2\) are both perpendicular to \(\mathbf{n}\), where \(\mathbf{n} = \begin{pmatrix} 1 \\ 2 \\ -2 \end{pmatrix}\). The points \(A(0, -9, 13)\) and \(B(8, 7, -3)\) lie in \(\Pi_1\) and \(\Pi_2\) respectively.
  1. Find the equations of \(\Pi_1\) and \(\Pi_2\) in the form \(\mathbf{r} \cdot \mathbf{n} = d\) and show that \(\overrightarrow{AB}\) is parallel to \(\mathbf{n}\). [4]
  2. Calculate the perpendicular distance between \(\Pi_1\) and \(\Pi_2\). [2]
  3. Write down two vectors which are perpendicular to \(\mathbf{n}\) and hence find, in the form $$\mathbf{r} = \mathbf{u} + \lambda\mathbf{v} + \mu\mathbf{w},$$ an equation for the plane \(\Pi_3\) which is parallel to \(\Pi_1\) and \(\Pi_2\) and exactly half-way between them. [4]
  4. The locus of all points \(P\) such that \(AP = BP = 12\sqrt{2}\) is denoted by \(L\).
    1. Give a full geometrical description of \(L\). [4]
    2. Using the result of part (iii), or otherwise, find a point on \(L\) which has integer coordinates. [4]