# Vector geometry - determinant proof

1. Nov 12, 2012

### bossman007

1. The problem statement, all variables and given/known data

Exercise 44 - In the picture attached

2. Relevant 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/ [Broken][/PLAIN]

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

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

3. 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

Last edited by a moderator: May 6, 2017
2. Nov 12, 2012

### tiny-tim

hi bossman007!
it should come out very simple

show us what you did

3. Nov 12, 2012

### bossman007

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

Last edited by a moderator: May 6, 2017
4. Nov 12, 2012

### tiny-tim

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

no, keep it simple

use the distributive law …

(C - B) x (B - A) … ?

5. Nov 12, 2012

### bossman007

many thanks :D,

what is the distributive law for 3 vectors?

6. Nov 12, 2012

### tiny-tim

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

7. Nov 12, 2012

### bossman007

thank you :D

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

8. Nov 12, 2012

### bossman007

is this right?

9. Nov 13, 2012

### tiny-tim

(just got up :zzz:)
nooo …

how did you get that result?

10. Dec 17, 2012

### boings

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

11. Dec 17, 2012

### HallsofIvy

Staff Emeritus
??? 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, $B\times B= 0$.