- The plane \(\Pi _ { 1 }\) has vector equation
$$\mathbf { r } \cdot ( 3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k } ) = 5$$
- Find the perpendicular distance from the point \(( 6,2,12 )\) to the plane \(\Pi _ { 1 }\)
The plane \(\Pi _ { 2 }\) has vector equation
$$\mathbf { r } = \lambda ( 2 \mathbf { i } + \mathbf { j } + 5 \mathbf { k } ) + \mu ( \mathbf { i } - \mathbf { j } - 2 \mathbf { k } )$$
where \(\lambda\) and \(\mu\) are scalar parameters.
- Show that the vector \(- \mathbf { i } - 3 \mathbf { j } + \mathbf { k }\) is perpendicular to \(\Pi _ { 2 }\)
- Show that the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) is \(52 ^ { \circ }\) to the nearest degree.