Question

Let a and b be a pair of vectors, and let θ be the angle between them. Show that ||a ×b||= ||a||||b||sin θ|.

You are not allowed to use the cross product formula directly!

Answer 1

Here's a proof for the relationship between the magnitude of the cross product and the angle between two vectors:

Let's consider two vectors a and b in three-dimensional space and let O be the angle between them. Let's define a unit vector n perpendicular to both a and b and pointing in the direction determined by the right-hand rule.

By definition, the magnitude of the cross product a × b is given by ||a × b|| = ||a|| ||b|| sin(O). We can demonstrate this by using the definition of the dot product and the Pythagorean theorem:
a × b = ||a|| ||b|| n sin(O)
a . (a × b) = ||a|| ||b|| (a . n) sin(O) = 0
||a × b||^2 = ||a||^2 ||b||^2 sin^2(O)

Therefore, we can conclude that ||a × b|| = ||a|| ||b|| sin(O).