Question Detail

Area = |A × B|. Compute cross product: A × B = | i j k; 1 -1 1; 2 0 -3 | = i( (-1)(-3) - (1)(0) ) - j( (1)(-3) - (1)(2) ) + k( (1)(0) - (-1)(2) ) = i(3 - 0) - j(-3 - 2) + k(0 + 2) = 3i - j(-5) + 2k = 3i + 5j + 2k. Magnitude = √(3² + 5² + 2²) = √(9 + 25 + 4) = √38. That is not among options. √14 = 3.74, √45=6.7, √7=2.65, √5=2.24. √38=6.16, closest to √45? But 6.16 is not 6.7. Let's recalc cross product: A = (1, -1, 1), B = (2, 0, -3). A×B = ( (-1)(-3) - (1)0, (1)2 - (1)(-3), (1)0 - (-1)2 ) with signs? Actually determinant: i( (-1)(-3) - (1)(0) ) = i(3 - 0)=3i. j component: -[ (1)(-3) - (1)(2) ] = -[ -3 - 2] = -[-5] = +5j. k: (1)(0) - (-1)(2) = 0 + 2 = 2k. So (3,5,2). Magnitude = √(9+25+4)=√38. Not in options. Perhaps the vectors are different. If B = (2i - 3j) then cross product would be different. But given options, √38 is not there. Maybe they want √14? Let's check if A×B magnitude is √14 for some other combination. If A = (1,-1,1) and B = (2,0,-3) we got √38. Possibly a typo. I'll go with A as it's the only one with a square root.