If `vec(a) = 3 hat(i) + hat(j) - 4 hat(k) , vec(b) = 6 hat(i) + 5 hat(j) - 2 hat(k)`. Find (a) ` vec(a) xx vec(b)` and (b) area of a triangle whose adjacent sides are dertermined by `vec(a)` and `vec(b)`.

Explanation

(a) `vec(a) xx vec(b) = |(+ , - , +),(hat(i),hat(j), hat(k)),(3 , 1 , -4) ,(6 , 5 , -2)|`
`= hat(i)|(1,-4), (5,-2)|- hat(j) |(3 , -4) , (6 , -2)| + hat(k)|( 3 , 1) , (6 , 5)|`
`= hat(i) {(1 xx - 2 ) - ( - 4 xx 5)} - hat(j) {(3 xx -2) - (-4 xx 6 )}+ hat (k) {( 3 xx 5 ) - ( 1 xx 6)}`
` = hat(i) { - 2 +20} - hat(j) { - 6 + 24} + hat(k) { 15 - 6}`
`= 18 hat(i) - 18 hat(j) + 9 hat(k)`
`| vec (a) xx vec(b)| = sqrt((18)^(2) + (-18)^(2) + (9)^(2)) = sqrt(729) = 27`
Area of triangle `= (1)/(2)| vec (a) xx vec(b)| = (27)/(2) = 13.5 unit`