If `alpha = 2i + 3j - k, beta = -i + 2j-4k, gamma = i+j+k` then the value of `(alpha xx beta).(alpha xx gamma)` is equal to

If bara = 2i + 3j - k , barb = - i + 2 j - 4k and barc = i + j + k . then (a xx b) · ( a xx c)

Let the line of the shortest distance between the lines L_1 : vec r = (hat i 2 hat j 3 hat k) lambda (hat i - hat j hat k) and L_2 : vec r = (4 hat i 5 hat j 6 hat k) mu (hat i hat j - hat k) intersect L_1 and L_2 at P and Q respectively. If (alpha, beta, gamma) is the mid point of the line segment PQ, then 2(alpha beta gamma) is equal to ___________.

Let vec alpha=2i+3j-k and vec beta=i+j. If vec gamma is a unit vector,then the maximum value of [vec alpha xxvec beta,vec beta xxvec gamma,vec gamma xxvec alpha] is equal to

If alpha + beta + gamma = pi , then the value of the determinant |(e^(2i alpha),e^(-i gamma),e^(-i beta)),(e^(-i gamma),e^(2i beta),e^(-i alpha)),(e^(-i beta),e^(-i alpha),e^(2i gamma))| , is

If alpha, beta, gamma are the roots of x^(3) + ax^(2) + b = 0 , then the value of |(alpha,beta,gamma),(beta,gamma,alpha),(gamma,alpha,beta)| , is

i(jxx k ) + j ( k xx i) + k (j xx i) is equal to