Find the shortest distance between the point \(( - 6,4 )\) and the line \(y = - 0.75 x + 7\).
Two lines, \(l _ { 1 }\) and \(l _ { 2 }\), are given by
\(l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } 4 3 - 2 \end{array} \right) + \lambda \left( \begin{array} { c } 2 1 - 4 \end{array} \right)\) and \(l _ { 2 } : \mathbf { r } = \left( \begin{array} { c } 11 - 1 5 \end{array} \right) + \mu \left( \begin{array} { c } 3 - 1 1 \end{array} \right)\).
Find the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\).
Hence determine the geometrical arrangement of \(l _ { 1 }\) and \(l _ { 2 }\).
Three matrices, \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\), are given by \(\mathbf { A } = \left( \begin{array} { c c } 1 & 2 a & - 1 \end{array} \right) , \mathbf { B } = \left( \begin{array} { c c } 2 & - 1 4 & 1 \end{array} \right)\) and \(\mathbf { C } = \left( \begin{array} { c c } 5 & 0 - 2 & 2 \end{array} \right)\) where \(a\) is a
constant.
Using \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) in that order demonstrate explicitly the associativity property of matrix multiplication.
Use \(\mathbf { A }\) and \(\mathbf { C }\) to disprove by counterexample the proposition 'Matrix multiplication is commutative'.
For a certain value of \(a , \mathbf { A } \binom { x } { y } = 3 \binom { x } { y }\).
Find
\(y\) in terms of \(x\),
the value of \(a\).
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The figure shows part of the graph of \(y = ( x - 3 ) \sqrt { \ln x }\). The portion of the graph below the \(x\)-axis is rotated by \(2 \pi\) radians around the \(x\)-axis to form a solid of revolution, \(S\).
Determine the exact volume of \(S\).