Math Help - Vector Orthogonality

1. Vector Orthogonality

Hey guys! I have a question which is pretty straightforward but I'm not sure how to prove it...

Show that for any vectors a, b, and c, vector v = b(a*c) - c(a*b) is orthogonal (perpendicular) to a

There is this theorem which states that if 3 vectors are coplanar, their triple product is equal to 0
Also, If a, b, and c are coplanar, then b * c will be orthogonal to a, hence their dot product will be equal to 0

That's all I could find and I'm not sure if I just prove it by theorem or if i have to do some arithmetic
Any help would be greatly appreciated!

2. Re: Vector Orthogonality

I think I know what question you're asking, but you should be careful with the * operation. Does * refer to a cross product or dot product? a*c would most likely be interpreted as a dot product, while $a \times c$ is clearly a cross product.

3. Re: Vector Orthogonality

I'm not too sure. In the question there is a period in between the letters which means to multiply
Edit: Looks like it is a dot product

4. Re: Vector Orthogonality

Originally Posted by hemster83
Show that for any vectors a, b, and c, vector $v = b(a\cdot c) - c(a\cdot b)$ is orthogonal (perpendicular) to a

There is this theorem which states that if 3 vectors are coplanar, their triple product is equal to 0
You don't need the triple product theorem. Just note that:
$(v\cdot a) = (b\cdot a)(a\cdot c) - (c\cdot a)(a\cdot b)=0~.$

5. Re: Vector Orthogonality

Originally Posted by Plato
You don't need the triple product theorem. Just note that:
$(v\cdot a) = (b\cdot a)(a\cdot c) - (c\cdot a)(a\cdot b)=0~.$
Thank you for the help!