# calculus

posted by .

show that if a, b & c are in v3, then (a cross b) dot [(b cross c)×(c cross a)] = [a dot (b cross c)]^2

• calculus -

If you look for some list of identities for vector products, such as wolfram: cross product you will find the following:

A.(BxC) = det(ABC)
(AxB)x(CxD) = det(ABD)C - det(ABC)D

If we adapt this formula to our problem, we need to set
A = b
B = c
C = c
D = a

Now, we plug and chug:
bxc x cxa = det(bca)c - det(bcc)a

Now, a determinant is zero if two rows are the same. So, we can toss out he last term on the right, and we are left with just

bxc x cxa = det(bca)c

AxB.C = C.AxB so,

axb . det(bca)c = det(bca)c . axb
= det(bca) * det(cab)

When we swap columns, the detrminant changes sign. Swapping columns twice, we get

det(bca) * det(cab) = det(abc) * det(abc) = (a.bxc)^2

## Respond to this Question

 First Name School Subject Your Answer

## Similar Questions

1. ### Math: Cross Product

Prove that | a x b | = (sqrt) [(a (dot) b)(b (dot) b) - (a (dot) b)^2]
2. ### MATH

FOUR CYCLISTS A,B,C&D WANTS TO CROSS A BRIDGE BUT DUE TO BAD CONDITION OF BRIDGE ONLY TWO AT A TIME CAN CROSS IT. IF A TAKES 1 MINUTES, B TAKES 2 MINUTES,C TAKES FIVE MINUTES & D TAKES 10 MINUTES TO CROSS THE BRIDGE, HOW THEY ALL CAN …
3. ### Chemistry

When drawing the 'dot and cross' diagram to show the ionic bonding in aluminium oxide, Al2O3, do we place the nos "2" and "3" in front of the brackets containing the 'dot and cross' diagram, e.g 2[Al]3[O]?
4. ### Chemistry

Hi Dr Bob, I understand the 2nd method but need to clarify the 1st method.If I only wish to draw 1 Al and 1 O, then I need to include the "plus" sign and draw the 'dot and cross' diagram as 2[Al]+ 3[O] , with the charges of course, …
5. ### calculus

Find two unit vectors that make an angle of 60° with v = ‹3, 4›. Give your answers correct to three decimal places. I'm not sure what to do here; I believe that I use to definition of a dot product (or maybe cross product) with …
6. ### Math

There are four people that want to cross a bridge. Sam can cross in 1 minute, Joe can cross in 2, Jack can cross in 5, and Steve can cross in 10. They have one flashlight, and two can cross at a time, but they must go at the same pace. …
7. ### maths magic

There are A,B,C,D at one end of the bridge.they have to cross within 17 minutes.only 2 can cross the bridge at atime .since it is very dark one has to come back to show the light and its said that A can cross the bridge in 1 min ,B …
8. ### Math

a) prove the identity: Ax(BXC)=B(A*C)-C(A*B) A,B,C vectors x- cross product *-dot product b) use part a) to prove that cross product is not associative.
9. ### math calculus

if i,j,k are the standard unit basic vectors, in 3 space, determine the value of k dot(j-3k)+(i-4k) dot (i-4k)-8 |i cross -k| (Simplify without using components)
10. ### Calculus

Find (a cross b)dot product a? & Find the cross product(a × b) dot product b?

More Similar Questions