Question

Find a set of the vectors of the reciprocal to the set: 3i - j + 2k , 2i + j - k and i - 2j + k

Answer 1

Given three non coplanar vectors a, b, c, the reciprocal vectors A, B, C are:
A = (b x c)/w
B = (c x a)/w
C = (a x b)/w
Where w = a . (b x c) is the volume of the parallelepiped fromed by a, b, and c.
Note: a x c means the cross product of the vectors a and c.
Now just plug the numbers in.

Answer 2

like i dont understand how we going to work with the fraction

Answer 3

do you know how to find the volume of the parallelepiped Mr._To mentioned ?
that is W in your formula...\[W=\vec a\cdot(\vec b\times\vec c)\]

Answer 4

guys please help...

Answer 5

what i know is that a . ( b x c ) is not equal to zero

Answer 6

well first you need\[\vec b\times\vec c\]do you know how to take the cross-product of two vectors?

Answer 7

yeah...

Answer 8

then that is step one

Answer 9

ohk let me work the answer and you tell if am wrong

Answer 10

let's say\[\vec a=3\hat i-\hat j+2\hat k\]\[\vec b=2\hat i+\hat j-\hat k\]\[\vec c=\hat i-2\hat j+\hat k\]so you tell me what \[\vec b\times\vec c\]is

Answer 11

-i - 3j - 5k

Answer 12

yeah, that's what I got :)
so what's\[\vec a\cdot(\vec b\times\vec c)\]?

Answer 13

10

Answer 14

-10 ain't it?

Answer 15

oh yeah -10

Answer 16

so now we know W=-10
you know how to take the cross-product of two vectors, so you should be able to use the formula @Mr._To provided

Answer 17

\[\vec A={\vec b\times\vec c\over W}\]\[\vec B={\vec c\times\vec a\over W}\]\[\vec C={\vec a\times\vec b\over W}\]

Answer 18

so in these quiz am looking for A B C

Answer 19

yup

Answer 20

So now cos the ( b x c ) is going to give me answer that got directions what am i suppose to do?

Answer 21

well \(\vec b\times\vec c\) is a vector, which is what we want
so now we just scalar divide that vector by W

Answer 22

which is the scalar?

Answer 23

W
scalar is a number without direction; basically the opposite of a vector

Answer 24

scalar multiplication on a vector is straightforward for each component

Answer 25

\[c\vec v=c\langle x,y,z\rangle=\langle cx,cy,cz\rangle\]

Answer 26

division works the same way of course

Answer 27

so i take the W to divide the vector?

Answer 28

yes, you are scalar multiplying each vector that you get from the cross-product by 1/W (which is the same thing as dividing by W....) to get the inverse

Answer 29

we already know \(\vec b\times\vec c\) from our earlier work
so now just scalar divide that by W

Answer 30

.... in order to find \(\vec A\)

Answer 31

lets say for example i go i - 2j + 5k i take w x i , w x (-2 )...

Answer 32

well the formula Mr._To has given is to divide by W, not multiply
but you could write it like that

Answer 33

ohk... so the answer suppose to be a vector or scalar?

Answer 34

better to be consistent in your notation; either\[\vec A=\frac1W(x\hat i+y\hat j+z\hat k)=\frac xW\hat i+\frac yW\hat j+\frac zW\hat k\]or\[\vec A=\frac1W\langle x,y,z\rangle=\langle \frac xW,\frac yW,\frac zW\rangle\]as you can see our answers are vectors

Answer 35

ohkay... i will try it out

Answer 36

Turing Test, great instructions!

Answer 37

Just following your lead Mr._To, that formula was news to me!