Vector product properties and identities

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
   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 (*).

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. [1 mark] \(\mathbf{a} \times (\mathbf{a} + \mathbf{b})\) \quad \((\mathbf{a} + \mathbf{b}) \times \mathbf{b}\) \quad \((\mathbf{a} - \mathbf{b}) \times \mathbf{b}\) \quad \(\mathbf{a} \times (\mathbf{a} - \mathbf{b})\)

The vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) are such that \(\mathbf{a} \times \mathbf{b} = \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}\) and \(\mathbf{a} \times \mathbf{c} = \begin{bmatrix} 0 \\ 0 \\ 3 \end{bmatrix}\) Work out \((\mathbf{a} - 4\mathbf{b} + 3\mathbf{c}) \times (2\mathbf{a})\) [4 marks]

Given that the vectors a and 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. [9 marks]

A transformation is equivalent to a shear parallel to the \(x\)-axis followed by a shear parallel to the \(y\)-axis and is represented by the matrix \(\begin{pmatrix} 1 & s \\ t & 0 \end{pmatrix}\). Find in terms of \(s\) the matrices which represent each of the shears. [7]