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Co-planarity help.

Okay I am doing my summer retakes and have no resources to check the methods for each question so I am going to post the question and my answer and please tell me whether I am right or completely wrong.

ma555 1.png

Here's what I've got:
1a) Let a.bxc represent the volume of the solid with vector sides a,b and c. If the volume is Zero, I.e a.bxc=0, then all three vectors must lie in the same plane. Therefore, if and only if a.bxc=0 are a,b and c coplanar.

b)

a=i+j+{\alpha}k, b=-i+2j+3k, c={\alpha}i+2{\alpha}j+9k

b\mathrm{x}c=(18-6{\alpha})i-(-9-3{\alpha})j+(-2{\alpha}-2{\alpha})k=(18-6{\alpha})i+(9+3{\alpha})j+(4\alpha)k

So:

a.b\mathrm{x}c=18-6{\alpha}+9+3{\alpha}-4{\alpha}^2=27-3{\alpha}-4{\alpha}^2

a.b\mathrm{x}c=0 \Longrightarrow 27-3{\alpha}-4{\alpha}^2=0 \Longrightarrow (4{\alpha}-9)({\alpha}+3)=0 \Longrightarrow {\alpha}=9/4 \mathrm{or} -3

Reply 1

ghostwalker

Original post by Happy2Guys1Hammer

...

Agree with part b, aside from the small typo.

Since part a is 6 of the 10 marks, I'd think that they'd want something more substantial, but it's years since I was at uni.

(edited 10 years ago)

Reply 2

Happy2Guys1Hammer

Original post by ghostwalker

Agree with part b, aside from the small typo.

Since part a is 6 of the 10 marks, I'd think that they'd want something more substantial, but it's years since I was at uni.

Would you say that I have shown that a,b and c are coplanar if and only if a.(bxc)=0? Cause I am not sure I have, I just did what I could.

Reply 3

ghostwalker

Original post by Happy2Guys1Hammer

Would you say that I have shown that a,b and c are coplanar if and only if a.(bxc)=0? Cause I am not sure I have, I just did what I could.

It's your opening line that is most disturbing "Let a.bxc represent the volume of the solid with vector sides a,b and c". I think you need to show that it does.

E.g. Consider the parallelopiped generated by the three vectors a,b,c which form three coincident sides. And then show that the volume is equal to a.bxc.

I suspect that there's as much work in that as there is in doing the whole thing from scratch with the definitions of the dot and vector product though.