Geometric interpretation of vector product

3 The non-zero vectors \(\mathbf { x }\) and \(\mathbf { y }\) are such that \(\mathbf { x } \times \mathbf { y } = \mathbf { 0 }\).

1. Explain the geometrical significance of this statement.
2. Use your answer to part (a) to explain how the line equation \(\mathbf { r } = \mathbf { a } + t \mathbf { d }\) can be written in the form \(( \mathbf { r } - \mathbf { a } ) \times \mathbf { d } = \mathbf { 0 }\).

4 Points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) respectively, relative to origin \(O\). It is given that \(\mathbf { b } \times \mathbf { c } = \mathbf { a }\) and that \(| \mathbf { a } | = 3\).

1. Determine each of the following as either a single vector or a scalar quantity.
   1. \(\mathbf { c } \times \mathbf { b }\)
   2. \(\mathbf { a } \times ( \mathbf { b } \times \mathbf { c } )\)
   3. \(\mathbf { a } \cdot ( \mathbf { b } \times \mathbf { c } )\)
2. Describe a geometrical relationship between the points \(O , A , B\) and \(C\) which can be deduced from
   1. the statement \(\mathbf { b } \times \mathbf { c } = \mathbf { a }\),
   2. the result of (a)(iii).

2

1. For non-zero vectors \(\mathbf { x }\) and \(\mathbf { y }\), explain the geometrical significance of the statement \(\mathbf { x } \times \mathbf { y } = \mathbf { 0 }\).
2. The points \(P\) and \(Q\) have position vectors \(\mathbf { p } = 2 \mathbf { i } + 7 \mathbf { j } - \mathbf { k }\) and \(\mathbf { q } = \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k }\) respectively. Find, in the form \(( \mathbf { r } - \mathbf { a } ) \times \mathbf { d } = \mathbf { 0 }\), the equation of line \(P Q\).