IF ` vecA xx vecB =vecC xx vecD and vecA xx vecC = vecB xx vecD + ` while ` | vecA| ne |vecD| | vecB| ne | vecC|` show `( vecA - vecD )` that and ` ( vecB - vecC)` are parallel

Explanation

Given
` vecA xx vecB = vecC xx vecD `
` vecA xx vecC = vecB xx vecD`
subtracting (2) from (1) we get
` vecA xx vecB - vecA xx vecB - vecA xx vecC = vecC xx vecD - vecB xx vecD`
` therefore vecA xx ( vecB - vecC )`
`= ( vecC - vecB ) xx vecD [ :. vecP xx ( vecQ +vecR) = vecP xx vecQ + vecP xx vecR]`
` therefore vecA xx ( vecB - vecC ) =- ( vecB - vecC ) xx vecD therefore vecA xx ( vecB - vecC ) xx vecD =0 therefore vecA xx vecB - vecC ) - vecD xx( vecB - vecC) =0`
`[ :. P vecA - vecD )xx ( vecB - vecC)=0`
Now `| vecA| ne | vecD| and | vecB | ne |vecC|`
` therefore ( vecA - vecD ) ne 0 and ( vecB - vecC ) ne 0`
`( vecA - vecD) and ( vecB - vecC)` are at an angle `0^@ or ( vecA - vecD) and ( vecB - vecC)` are parallel
`[ ,. vecP xx vecQ =PQ sin theta and P ne 0 ne Q ` hence sin ` theta =0`]