Question

If a, b, c and d lie in the same plane when positioned with a common initial point, then prove that (axb)×(cxd)=0​

Answer 1

Step-by-step explanation:

The way I remember it is that (a⃗ ×b⃗ ) will produce a vector v⃗ which will be orthogonal to the vectors a⃗ and b⃗ , which in this case will be perpendicular to the plane.

However in this case you also have (c⃗ ×d⃗ )=u⃗ which would also be perpendicular to the plane.

Therefore (u⃗ ×v⃗ )=∥u∥∥v∥sin(θ), since u⃗ and v⃗ are perpendicular to the plane, you'll either have θ=0 or θ=π, showing that u⃗ and v⃗ are either parallel or in opposite directions. So in the end you ll finally have (a⃗ ×b⃗ )×(c⃗ ×d⃗ )=u⃗ ×v⃗ =∥u∥∥v∥sin(θ)=0

Answer 2

Step-by-step explanation:

If a, b, c, and d lie in the same plane, then (a×b)×(c×d)=0

I can't really see a way to prove this other than

if (a×b) is an orthogonal vector v in Rn and (c×d) is an orthogonal vector u in Rn then cross product of u×v has to lie in the original plane as the cross product is perpendicular to both planes

As you can see this isn't very general or formal for that matter.

pls mark me on brainlist