Vector geometry - determinant proof

[bossman007]

Homework Statement

Exercise 44 - In the picture attached

Homework Equations

HINT - expand the expression for n and plug the result into equation (70), then use equation (63)

n=(C-B)X(B-A)

n dot (r - A) = 0 (eq. 70)

A dot (B X C) = det {A B C} (eq. 63)

[PLAIN]http://postimage.org/image/68xx59akh/ [/PLAIN]

[PLAIN]http://postimage.org/image/rzbcga401/ [/PLAIN]

[PLAIN]http://postimage.org/image/wtauu6jv9/ [/PLAIN]

The Attempt at a Solution

Like the hint said, I expanded the expression for n , which is (C-B)X(B-A) and I got a giant mess. I have no idea if I did it right or am doing it right or what to do here

 

[tiny-tim]

hi bossman007!

Like the hint said, I expanded the expression for n , which is (C-B)X(B-A) and I got a giant mess.

it should come out very simple …

show us what you did ​

 

[bossman007]

[PLAIN]http://postimage.org/image/j5wfm5rjr/ [/PLAIN]

 

[tiny-tim]

ohhh! that's why it's a giant mess!

no, keep it simple …

use the distributive law …

(C - B) x (B - A) … ? ​

 

[bossman007]

many thanks :D,

what is the distributive law for 3 vectors?

 

[tiny-tim]

P x (Q + R) = (P x Q) + (P x R)

 

[bossman007]

thank you :D

so that means in my case I do B x (C-A) ?

 

[bossman007]

is this right?

 

[tiny-tim]

(just got up :zzz:)

(C - B) x (B - A) … ? ​

thank you :D

so that means in my case I do B x (C-A) ?

nooo …

how did you get that result? ​

 

[boings]

What should this be exactly? I'm not sure I understand either

 

[HallsofIvy]

thank you :D

so that means in my case I do B x (C-A) ?

? There is NO "C- A" in the expression you want to expand!

(C- B)\times (B- A)= C\times(B- A)- B\times(B- A)
That's one step, now do it again:
C\times B- C\times A- B\times B+ B\times A
and, of course, B\times B= 0.