Answer
a) $v \times w =-8i+5j +14k$
b) $w \times v=8i-5j-14k$
c) $ v \times v =0$
d) $ w \times w =0$
Work Step by Step
Let us consider two vectors $v=xi+yj+zk$ and $w=pi+qj+rk$, then cross product of two vectors $v$ and $w$ can be computed in the form of determinate as : $ v \times w=\begin{vmatrix} i & j & k \\ x & y & z \\ p & q & r \\ \end{vmatrix}$Supposes that the two vectors can be represented as: $v=xi+yj+zk$ and $w=pi+qj+rk$, then their cross product of such vectors can be obtained in the form of determinate as :
$ v \times w=\begin{vmatrix} i & j & k \\ x & y & z \\ p & q & r \\ \end{vmatrix}$
a) $v \times w =[(-4)(1)-(2)(2)] i -j [(1)(1) - (2)(3)]+k [(1)(2) -(-4) (3)]=-8i+5j +14k$
b) Since, a cross or vector product is not commutative. So we can write as: $v \times w= -w \times v$
So, $w \times v=8i-5j-14k$
c) We know that for the two mutually perpendicular vectors, we have: $v \times v = 0$
d) We know that for the two mutually perpendicular vectors, we have: $w \times w =0$
