Vector product properties and identities

14 Given that the vectors \(\mathbf { a }\) and \(\mathbf { b }\) are perpendicular, prove that \(| ( \mathbf { a } + 5 \mathbf { b } ) \times ( \mathbf { a } - 4 \mathbf { b } ) | = k | \mathbf { a } | | \mathbf { b } |\), where \(k\) is an integer to be found. Explicitly state any properties of the vector product that you use within your proof.
[0pt] [9 marks] LL

1 The points \(A , B\) and \(C\) have position vectors \(\mathbf { a } = \left( \begin{array} { l } 1 \\ 1 \\ 3 \end{array} \right) , \mathbf { b } = \left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right)\) and \(\mathbf { c } = \left( \begin{array} { r } - 5 \\ 1 \\ 2 \end{array} \right)\) respectively, relative to the origin \(O\).

1. Calculate, in its simplest exact form, the area of triangle \(O A B\).
2. Show that \(\mathbf { a } \times ( \mathbf { b } \times \mathbf { c } ) + \mathbf { b } \times ( \mathbf { c } \times \mathbf { a } ) + \mathbf { c } \times ( \mathbf { a } \times \mathbf { b } ) = \mathbf { 0 }\).

6

1. For the vectors \(\mathbf { p } = \left( \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right) , \mathbf { q } = \left( \begin{array} { r } 3 \\ 1 \\ - 1 \end{array} \right)\) and \(\mathbf { r } = \left( \begin{array} { r } 2 \\ - 4 \\ 5 \end{array} \right)\), calculate
   • \(\mathbf { p } \cdot \mathbf { q } \times \mathbf { r }\),
   • \(\mathbf { p } \times ( \mathbf { q } \times \mathbf { r } )\),
   • \(( \mathbf { p } \times \mathbf { q } ) \times \mathbf { r }\).
   • State whether the vector product is associative for three-dimensional column vectors with real components. Justify your answer.
   It is given that \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) are three-dimensional column vectors with real components.
2. Explain geometrically why the vector \(\mathbf { a } \times ( \mathbf { b } \times \mathbf { c } )\) must be expressible in the form \(\lambda \mathbf { b } + \mu \mathbf { c }\), where \(\lambda\) and \(\mu\) are scalar constants. It is given that the following relationship holds for \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\). \(\mathbf { a } \times ( \mathbf { b } \times \mathbf { c } ) = ( \mathbf { a } \cdot \mathbf { c } ) \mathbf { b } - ( \mathbf { a } \cdot \mathbf { b } ) \mathbf { c }\)
3. Find an expression for ( \(\mathbf { a } \times \mathbf { b ) } \times \mathbf { c }\) in the form of (*).

1 Three of the four expressions below are equivalent to each other.
Which of the four expressions is not equivalent to any of the others? Circle your answer. \(\mathbf { a } \times ( \mathbf { a } + \mathbf { b } )\) \(( \mathbf { a } + \mathbf { b } ) \times \mathbf { b }\) \(( \mathbf { a } - \mathbf { b } ) \times \mathbf { b }\) \(\mathbf { a } \times ( \mathbf { a } - \mathbf { b } )\)