8.04c Areas using vector product: triangles and parallelograms

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 (*).

5 You are given the variable point \(A ( 3 , - 8 , t )\), where \(t\) is a real parameter, and the fixed point \(B ( 1,2 , - 2 )\).

1. Using only the geometrical properties of the vector product, explain why the statement " \(\overrightarrow { \mathrm { OA } } \times \overrightarrow { \mathrm { OB } } = \mathbf { 0 }\) " is false for all values of \(t\).
2. 1. Use the vector product to find an expression, in terms of \(t\), for the area of triangle \(O A B\).
   2. Hence determine the value of \(t\) for which the area of triangle \(O A B\) is a minimum.

5 The points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) respectively, relative to a fixed origin \(O\).

1. (a) Prove that \(\mathbf { a } \times ( \mathbf { b } - \mathbf { a } ) = \mathbf { a } \times \mathbf { b }\).
   (b) Determine the relationship between \(\mathbf { a } \times ( \mathbf { b } - \mathbf { a } )\) and \(\mathbf { b } \times ( \mathbf { b } - \mathbf { a } )\).
2. The point \(D\) is on the line \(A B\). \(O D\) is perpendicular to \(A B\). By considering the area of triangle \(O A B\), show
   that \(| O D | = \frac { | \mathbf { a } \times \mathbf { b } | } { | \mathbf { b } - \mathbf { a } | }\).

5 Torque is a vector quantity that measures how much a force acting on an object causes that object to rotate. The torque (about the origin), \(\mathbf { T }\), exerted on an object is given by \(\mathbf { T } = \mathbf { p } \times \mathbf { F }\), where \(\mathbf { F }\) is the force acting on the object and \(\mathbf { p }\) is the position vector of the point at which \(\mathbf { F }\) is applied to the object. The points \(A\) and \(B\), with position vectors \(\mathbf { a } = 3 \mathbf { i } + 4 \mathbf { j } + \mathbf { k }\) and \(\mathbf { b } = 3 \mathbf { i } + 5 \mathbf { j } + \mathbf { k }\) are on the surface of a rock. The force \(\mathbf { F } _ { 1 } = 6 \mathbf { i } + 7 \mathbf { j } - 3 \mathbf { k }\) is applied to the rock at \(A\) while the force \(\mathbf { F } _ { 2 } = - 7 \mathbf { i } - 10 \mathbf { j } + 2 \mathbf { k }\) is applied to the rock at \(B\).

1. Find the torque (about the origin) exerted on the rock by \(\mathbf { F } _ { 1 }\).
2. Determine which of the two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) exerts a torque (about the origin) of greater magnitude on the rock.
3. Show that the torque (about the origin) is the same as your answer to part (a) when \(\mathbf { F } _ { 1 }\) acts on the rock at any point on the line \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { p }\), where \(\mathbf { p }\) is a vector in the same direction as \(\mathbf { F } _ { 1 }\). A third force \(\mathbf { F } _ { 3 }\) is now applied to the rock at \(A\), which exerts zero torque (about the origin).
4. Show that \(\mathbf { F } _ { 3 }\) must act in the direction of the line through \(A\) and the origin.

2 The points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) relative to the origin \(O\). It is given that \(\mathbf { a } = \left( \begin{array} { c } 2 \\ 4 \\ 3 \lambda \end{array} \right)\) and \(\mathbf { b } = \left( \begin{array} { r } \lambda \\ - 4 \\ 6 \end{array} \right)\), where \(\lambda\) is a real parameter.

1. In the case when \(\lambda = 3\), determine the area of triangle \(O A B\).
2. Determine the value of \(\lambda\) for which \(\mathbf { a } \times \mathbf { b } = \mathbf { 0 }\).

1 The points \(A , B\) and \(C\) have position vectors \(\mathbf { a } = \left( \begin{array} { l } 3 \\ 0 \\ 0 \end{array} \right) , \mathbf { b } = \left( \begin{array} { l } 0 \\ 4 \\ 0 \end{array} \right)\) and \(\mathbf { c } = \left( \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right)\) respectively, relative to the origin \(O\).

1. 1. Calculate \(\mathbf { a } \times \mathbf { b }\), giving your answer as a multiple of \(\mathbf { c }\).
   2. Explain, geometrically, why \(\mathbf { a } \times \mathbf { b }\) must be a multiple of \(\mathbf { c }\).
2. Use a vector product method to calculate the area of triangle \(A B C\).