Vector equation of a line

8 The line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { r } 1 \\ 2 \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { l } 2 \\ 1 \\ 3 \end{array} \right)\). The plane \(p\) has equation \(\mathbf { r } \cdot \left( \begin{array} { r } 2 \\ - 1 \\ - 1 \end{array} \right) = 6\).

1. Show that \(l\) is parallel to \(p\).
2. A line \(m\) lies in the plane \(p\) and is perpendicular to \(l\). The line \(m\) passes through the point with coordinates (5, 3, 1). Find a vector equation for \(m\).

3 Two fixed points, \(A\) and \(B\), have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) relative to the origin \(O\), and a variable point \(P\) has position vector \(\mathbf { r }\).

1. Give a geometrical description of the locus of \(P\) when \(\mathbf { r }\) satisfies the equation \(\mathbf { r } = \lambda \mathbf { a }\), where \(0 \leqslant \lambda \leqslant 1\).
2. Given that \(P\) is a point on the line \(A B\), use a property of the vector product to explain why \(( \mathbf { r } - \mathbf { a } ) \times ( \mathbf { r } - \mathbf { b } ) = \mathbf { 0 }\).
3. Give a geometrical description of the locus of \(P\) when \(\mathbf { r }\) satisfies the equation \(\mathbf { r } \times ( \mathbf { a } - \mathbf { b } ) = \mathbf { 0 }\).

1

1. Find a vector which is perpendicular to both \(\left( \begin{array} { r } 1 \\ 3 \\ - 2 \end{array} \right)\) and \(\left( \begin{array} { r } - 3 \\ - 6 \\ 4 \end{array} \right)\).
2. The cartesian equation of a line is \(\frac { x } { 2 } = y - 3 = 2 z + 4\). Express the equation of this line in vector form.