Two questions on vectors, regarding dot and cross product?

[Jormungandr]

Homework Statement

1. Suppose that u + v + w = 0. Show that u x v = v x w = w x u. What is the geometric interpretation of this result? (Note: The interpretation should explain both the length and the direction).

2. Let v1, v2, and v be three mutually orthogonal vectors in space. Use the dot product to show that if c1, c2, and c3 are scalars such that c1v1 + c2v2 + c3v3 = 0, then c1 = c2 = c3 = 0.

Homework Equations

The Attempt at a Solution

1. This one was largely easy enough. I just made three vectors that added up to 0, crossed each of them with each other and sure enough I got the same vector for each. I repeated with another, and it's easily verifiable. The problem comes with the interpretation.
I realized that the product vector must be a vector that is orthogonal to all three original vectors, so it follows that the only logical conclusion there must be is that the vectors u, v, and w are on the same plane. What does this have to do with the length, though? The length of the resultant vector is, as far as I can tell, unrelated to those of the original three vectors. The equation relating the magnitude of the cross product and the sine of the angle between the vectors also yields no meaningful result.

2. This one makes sense intuitively, but I can't think of a mathematical way to prove it. If the three vectors are mutually orthogonal, then they obviously can't lie on the same plane. And the only way three vectors add up to 0 are if they're on the same plane, right? That is, unless the scalars are 0 themselves, which is the only other way I can see them adding up to 0. So yes, it makes sense if you think about it, but I can't really write that down.

 

[LCKurtz]

Homework Statement

2. Let v1, v2, and v be three mutually orthogonal vectors in space. Use the dot product to show that if c1, c2, and c3 are scalars such that c1v1 + c2v2 + c3v3 = 0, then c1 = c2 = c3 = 0.

2. This one makes sense intuitively, but I can't think of a mathematical way to prove it. If the three vectors are mutually orthogonal, then they obviously can't lie on the same plane. And the only way three vectors add up to 0 are if they're on the same plane, right? That is, unless the scalars are 0 themselves, which is the only other way I can see them adding up to 0. So yes, it makes sense if you think about it, but I can't really write that down.

Hint: What happens if you dot $\vec v_1$ into both sides of your given equation?

 

[Jormungandr]

Hint: What happens if you dot $\vec v_1$ into both sides of your given equation?

That was brilliant! Splendid hint, it was all I needed to get the answer, thank you!
Just to confirm, dotting $\vec v_1$ into both sides left me with c1|$\vec v_1$|² = 0, which of course implies that c1 = 0. I repeated by dotting $\vec v_2$ into the original, and by the same process arrived at c2 = 0. Repeating by dotting $\vec v_3$ or by substituting c1 = c2 = 0 into the original given equation, we finally are able to arrive at c3 = 0. Rather clean and a very nice solution. Thank you once again.

 

[LCKurtz]

That was brilliant! Splendid hint, it was all I needed to get the answer, thank you!
Just to confirm, dotting $\vec v_1$ into both sides left me with c1|$\vec v_1$|² = 0, which of course implies that c1 = 0. I repeated by dotting $\vec v_2$ into the original, and by the same process arrived at c2 = 0. Repeating by dotting $\vec v_3$ or by substituting c1 = c2 = 0 into the original given equation, we finally are able to arrive at c3 = 0. Rather clean and a very nice solution. Thank you once again.

Right. I call that taking the hint and running with it. Good job.

 

[Dick]

For the first one, express w in terms of u and v. Then substitute that in the cross product equation. For a geometric interpretation try to show that they are three different ways of finding a signed area of the same triangle.