Question

If A,B,C are any three vectors , show that A X (B+C)=AXB+AXC.

Answer 1

let

A = 1 i + 2 j

B = 3 i + 4 j

C = 5 i + 6 j

L.H.S

A x (B + C)

(1 i + 2 j ) x ((3 i + 4 j) + (5 i + 6 j ))

(1 i + 2 j ) x (8 i + 10 j)

10 k - 16 k

- 6 k

R.H.S

A x B + A x C

((1 i + 2 j) x (3 i + 4 j)) + ((1 i + 2 j) x (5 i + 6 j))

(4 k - 6 k) + (6 k - 10 k)

- 6 k

SInce L.H.S = R.H.S

hence proved

Answer 2

Let A = Axi + Ayj + Azk

B = Bxi + Byj + Bzk

C = Cxi + Cyj + Cxk

Adding B and C, you will get Z = i(Bx +Cx) + j(By +Cy) + k( Bz+ Cz)

Now applying cross product to A and Z = B + C,

You will get i(Ay(Bz +Cz) -Az(By +Cy) +j........ accordingly..

Then Apply cross product to A and B , and A and C and add them.

You will get osmething like this; i(AyBz -AzBy +AyCz -AzCy) + j.......

Taking Ay and Az common, you'll get i(Ay(Bz+Cz) -Az(By+Cy)) + j.....

Solve it in this way and you will get your result.