Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

a) $v \times w =3i-j +4k$ b)$w \times v=-3i+j-4k$ c) $v \times v =0$ d) $w \times w =0$
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) $v \times w =[(-1)(-3)-(-1)(0)] i -j [(1)(-3) - (-1)(4)]+k [(1)(0) -(-1) (4)]=3i-j +4k$ b) Since, a cross or vector product is not commutative. So we can write as: $v \times w= -w \times v$ So, $w \times v=-3i+j-4k$ 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: $w \times w =0$