Point on line with condition

Questions asking to find a specific point on a line satisfying a geometric condition such as perpendicularity to another vector, equal distances, or a given angle.

26 questions · Standard +0.5

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AQA C4 2008 June Q7
12 marks Standard +0.8
7 The coordinates of the points \(A\) and \(B\) are ( \(3 , - 2,1\) ) and ( \(5,3,0\) ) respectively. The line \(l\) has equation \(\mathbf { r } = \left[ \begin{array} { l } 5 \\ 3 \\ 0 \end{array} \right] + \lambda \left[ \begin{array} { r } 1 \\ 0 \\ - 3 \end{array} \right]\).
  1. Find the distance between \(A\) and \(B\).
  2. Find the acute angle between the lines \(A B\) and \(l\). Give your answer to the nearest degree.
  3. The points \(B\) and \(C\) lie on \(l\) such that the distance \(A C\) is equal to the distance \(A B\). Find the coordinates of \(C\).