[Math] What are the relationships between the three vectors in $\vec a • (\vec b × \vec c)$

vectors

I'm given this task:

Given a = [-5, 8, 1] and b = [2, -7, -3], find

  1. a × b and verify that it is perpendicular to both a and b

  2. Avector c such that a • (b × c) = 0

What is the relationship between the vectors a, b and c in this case, and why? Verify this.

I have already done the points 1 and 2, but I'm not sure about my answer to the "What is the relationship between the vectors a, b and c in this case, and why? Verify this."

So far I've found that a and c have to be collinear, but otherwise I see no other relationships. However it feels like there should be more, so am I missing something?

Best Answer

$a\cdot(b \times c)=(a \times b) \cdot c=0$

This implies $c$ is perpendicular to $a \times b$.

So what's the relation with $c$ and $a,b$ ? (Note: $a \times b$ is perpendicular to both $a$ and $b$).

Do you observe that c is parallel to the plane spanned by a and b?

therefore c= xa+yb for some x,y in R.

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