Question

Let the vectors `veca, vecb, vecc` given as `a_1hati + a_2hatj + a_3hatk, b_1hati + b_2hatj + b_3hatk, c_1hati + c_2hatj + c_3hatk` Then show that = `veca xx (vecb+ vecc) = veca xx vecb + veca xx vecc.`

Answer 1

Given `a_1 hati + a_2hatj + a_3hatk, b_1hati + b_2hatj + b_3hatk, c_1hati + c_2hatj + c_3hatk`

To prove that `veca xx (vecb + vecc) = veca xx vecb + veca xx vecc`

`veca = a_1hati + a_2hatj + a_3hatk`

`vecb = b_1hati + b_2hatj + b_3hatk`

`vecc = c_1hati + c_2hatj + c_3hatk`

`(vecb + vecc) = (b_1 + c_1)hati + (b_2 + c_2)hatj + (b_3 + c_3)hatk`

`veca xx (vecb + vecc) = |(hati,  hatj,  hatk),(a_1,  a_2,  a_3), (b_1 + c_1, b_2 + c_2, b_3 + c_3)|`

`= hati [a_2b_3 + a_2c_3 - a_3b_2 - a_3c_2] + hatj [- a_1b_3 - a_1c_3 + a_3b_1 + a_3c_1] + hatk [a_1b_2 + a_1c_2 - a_2b_1 - a_2c_1]`       ....(i)

`veca xx vecb = |(hati, hatj, hatk), (a_1, a_2, a_3), (b_1, b_2, b_3)|`

`= hati [a_2b_3 - a_3b_2] + hatj [a_1b_3 - a_3b_1] + hatk [a_1b_2 - a_2b_1]`        ...(ii)

`veca xx vecc = |(hati, hatj, hatk), (a_1, a_2, a_3), (c_1, c_2, c_3)|`

`= hati [a_2c_3 - a_3c_2] + hatj [a_3c_1 - a_1 c] + hatk [a_1c_2 - a_2c_1]`        ...(iii)

Adding (ii) and (iii)

`hati [a_2c_3 - a_3c_2 + a_2b_3 - a_3b_2] + hatj [a_3c_1 - a_1c + a_1b_3 - a_3b_1] + hatk [a_1c_2 - a_2c_1 + a_1b_2 - a_2b_1]`   ....(iv)

From (i) and (iv)

`veca xx (vecb + vecc) = veca xx vecb + veca xx vecc`