Question

Find \(|u \times v|\) and determine whether \(u \times v\) is directed into the page or out of the page.

Short answer

\(|u \times v|\) is \(10\sqrt 2 \) and \(u \times v\) is directed out of the page.

Step-by-step solution

Formula used

If\(\theta \)is the angle between vectors\({\bf{u}}\)and\({\bf{v}}\), then the cross product is,

\(|u \times v| = |u||v|\sin \theta \ldots \ldots \ldots (1)\)

Here,

|u| is the magnitude of vector\({\bf{u}}\)and

|v| is the magnitude of vector\({\bf{v}}\).

Right-hand rule:

The fingers of the right hand curl in the direction of rotation from\({\bf{u}}\)to\({\bf{v}}\), then the thumb points are in the direction of\(u \times v\). The rotation angle should be less than\({180^^\circ }\).

Find the \(|u \times v|\)

As given, \(|{\rm{u}}| = 4,|{\rm{v}}| = 5\) and\(\theta = {45^^\circ }\).

Substitute 4 for |u|, 5 for \(|{\rm{v}}|\) and \({45^^\circ }\) for \(\theta \) in equation (1),

\(\begin{array}{l}|{\rm{u}} \times {\rm{v}}| = (4)(5)\sin \left( {{{45}^^\circ }} \right)\\|{\rm{u}} \times {\rm{v}}| = 20\left( {\frac{{\sqrt 2 }}{2}} \right)\\|{\rm{u}} \times {\rm{v}}| = 10\sqrt 2 \end{array}\)

Thus, the \(|{\rm{u}} \times {\rm{v}}|\)is\(10\sqrt 2 \).

Since, the rotation angle is\({45^^\circ }\), which is from \(u\) to \(v\), the resultant \(u \times v\) is directed out of the page (by the right-hand rule).

Thus, the resultant \(u \times v\) is directed out of the page.