Let ` vec a= hat i+4 hat j+2 hat k ,\ \ vec b=3 hat i-\ 2 hat j+7 hat k` and ` vec c=2 hat i-\ hat j+4 hat k` . Find a vector ` vec d` which is perpendicular to both ` vec a` and ` vec b` and ` vec c` . ` vec d` =15.

Explanation

`implies (2hat(i) - hat(j) + 4hat(k)).lambda(32hat(i)-hat(j)-14hat(k)) = 15`
`implies 2 xx (32lamda) + (-1) xx(-lambda) + 4 xx (-14lambda) = 15`
`implies 64 lambda + lambda - 56 lambda = 15`
`implies 9lambda = 15 implies lambda = (15)/(9) = (5)/(3)`
`therefore` Required vector is `vec(d) = (5)/(3) (32hat(i) - hat(j) - 14hat(k))`