Prove that A.(A cross B) =0

Asked by JAIBHARGAVA200 | 13th Oct, 2014, 02:24: PM

Expert Answer:

begin mathsize 14px style Consider space two space vectors space straight A with rightwards arrow on top space and space straight B with rightwards arrow on top The space vectors space can space be space written space in space their space rectangular space components space as : straight A with rightwards arrow on top equals straight A subscript straight x straight i with overparenthesis on top space plus straight A subscript straight y space straight j with overparenthesis on top space plus straight A subscript straight z space straight k with overparenthesis on top straight B with rightwards arrow on top equals straight B subscript straight x straight i with overparenthesis on top space plus straight B subscript straight y space straight j with overparenthesis on top space plus straight B subscript straight z space straight k with overparenthesis on top straight A with rightwards arrow on top cross times straight B with rightwards arrow on top space equals open vertical bar table row cell straight i with overparenthesis on top end cell cell straight j with overparenthesis on top end cell cell straight k with overparenthesis on top end cell row cell straight A subscript straight x end cell cell straight A subscript straight y end cell cell straight A subscript straight z end cell row cell straight B subscript straight x end cell cell straight B subscript straight y end cell cell straight B subscript straight z end cell end table close vertical bar space space space space space space space space space space equals space straight i with overparenthesis on top space open parentheses straight A subscript straight y straight B subscript straight z minus straight A subscript straight z straight B subscript straight y close parentheses minus straight j with overparenthesis on top space open parentheses straight A subscript straight x straight B subscript straight z minus straight A subscript straight z straight B subscript straight x close parentheses plus straight k with overparenthesis on top space open parentheses straight A subscript straight x straight B subscript straight y minus straight A subscript straight y straight B subscript straight x close parentheses straight A with rightwards arrow on top. open parentheses straight A with rightwards arrow on top cross times straight B with rightwards arrow on top close parentheses equals open parentheses straight A subscript straight x straight i with overparenthesis on top space plus straight A subscript straight y space straight j with overparenthesis on top space plus straight A subscript straight z space straight k with overparenthesis on top close parentheses space space. space space open parentheses straight i with overparenthesis on top space open parentheses straight A subscript straight y straight B subscript straight z minus straight A subscript straight z straight B subscript straight y close parentheses minus straight j with overparenthesis on top space open parentheses straight A subscript straight x straight B subscript straight z minus straight A subscript straight z straight B subscript straight x close parentheses plus straight k with overparenthesis on top space open parentheses straight A subscript straight x straight B subscript straight y minus straight A subscript straight y straight B subscript straight x close parentheses close parentheses space space space space space space space space space space space space space space space space equals space straight A subscript straight x open parentheses straight A subscript straight y straight B subscript straight z minus straight A subscript straight z straight B subscript straight y close parentheses space minus straight A subscript straight y open parentheses straight A subscript straight x straight B subscript straight z minus straight A subscript straight z straight B subscript straight x close parentheses plus straight A subscript straight z space open parentheses straight A subscript straight x straight B subscript straight y minus straight A subscript straight y straight B subscript straight x close parentheses  open parentheses straight W space straight e space know space that space straight i with overparenthesis on top. straight i with overparenthesis on top space equals straight j with overparenthesis on top. straight j with overparenthesis on top space equals straight k with overparenthesis on top. straight k with overparenthesis on top space equals space 1 close parentheses  rightwards double arrow space straight A with rightwards arrow on top. open parentheses straight A with rightwards arrow on top cross times straight B with rightwards arrow on top close parentheses space equals straight A subscript straight x straight A subscript straight y straight B subscript straight z minus straight A subscript straight x straight A subscript straight z straight B subscript straight y minus straight A subscript straight y straight A subscript straight x straight B subscript straight z minus straight A subscript straight y straight A subscript straight z straight B subscript straight x plus straight A subscript straight z straight A subscript straight x straight B subscript straight y minus straight A subscript straight z straight A subscript straight y straight B subscript straight x space space space space space space space space space space space space space space space space space space space space space equals 0 rightwards double arrow space straight A with rightwards arrow on top. open parentheses straight A with rightwards arrow on top cross times straight B with rightwards arrow on top close parentheses equals 0 end style
Thus proved.

Answered by Jyothi Nair | 14th Oct, 2014, 09:09: AM