OCR
Further Additional Pure
2019
June
— Question 6
Exam Board
OCR
Module
Further Additional Pure (Further Additional Pure)
Year
2019
Session
June
Topic
Vector Product and Surfaces
6
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.
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 \(\mathbf { c }\).
\(\mathbf { a } \times ( \mathbf { b } \times \mathbf { c } ) = ( \mathbf { a } \cdot \mathbf { c } ) \mathbf { b } - ( \mathbf { a } \cdot \mathbf { b } ) \mathbf { c }\)
Find an expression for ( \(\mathbf { a } \times \mathbf { b ) } \times \mathbf { c }\) in the form of (*).