Looks like you're given
a = 2 i + 3 j + 4 k
b = i - j + k
Recall the properties of the cross product:
i x i = j x j = k x k = 0
i x j = k
j x k = i
k x i = j
and for any two vectors u and v,
u x v = -(v x u)
(i) Compute the cross product:
a x b = (2 i + 3 j + 4 k) x (i - j + k)
a x b = 2 (i x i) + 3 (j x i) + 4 (k x i)
.......... - 2 (i x j) - 3 (j x j) - 4 (k x j)
.......... + 2 (i x k) + 3 (j x k) + 4 (k x k)
a x b = (3 + 4) (j x k) + (4 - 2) (k x i) + (-2 - 3) (i x j)
a x b = 7 i + 2 j - 5 k
(ii) Recall the formula,
||a x b|| = ||a|| ||b|| sin(θ)
where θ is the angle between the vectors a and b.
We have
||a x b|| = √(7^2 + 2^2 + (-5)^2) = √78
||a|| = √(2^2 + 3^2 + 4^2) = √29
||b|| = √(1^2 + (-1)^2 + 1^2) = √3
so that
sin(θ) = √(78/87)
(iii) The cross product a x b is already perpendicular to both a and b, so you just need to normalize this vector. This is done by dividing it by its magnitude:
(a x b)/||a x b|| = 7/√78 i + 2/√78 j - 5/√78 k