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.