Question

find (a) v $\times$w, (b) w $\times$v, (c) w $\times$w, and

(d) v $\times$v.

v= 2i - 3j + k

w = 3i - 2j - k

Short answer

v ×w= 5i+5j+5k

w ×v= -5i-5j-5k

w ×w=0

v ×v=0

Step-by-step solution

Step 1. Given information 

v= 2i - 3j + k

w = 3i - 2j - k

Step 2. v×w

For two vectors

v=a₁i+b₁j+c₁k

w=a₂i+b₂j+c₂k

cross product v$\times$w is written as

$v\times w=\left|\begin{array}{ccc}i& j& k\\ a_1& b_1& c_1\\ a_2& b_2& c_2\end{array}\right|=i\left|\begin{array}{cc}{b}_1& c_1\\ b_2& c_2\end{array}\right|-j\left|\begin{array}{cc}{a}_1& c_1\\ a_2& c_2\end{array}\right|+k\left|\begin{array}{cc}{a}_1& b_1\\ a_2& b_2\end{array}\right|$

v= 2i - 3j + k

w = 3i - 2j - k

$$\begin{gathered} v\times w=\left|\begin{array}{ccc}i& j& k\\ 2& -3& 1\\ 3& -2& -1\end{array}\right| \\ =i\left|\begin{array}{cc}-3& 1\\ -2& -1\end{array}\right|-j\left|\begin{array}{cc}2& 1\\ 3& -1\end{array}\right|+k\left|\begin{array}{cc}2& -3\\ 3& -2\end{array}\right| \\ =5i+5j+5k \end{gathered}$$

Step 3. w ×v

According to the algebraic properties of cross product w$\times$v=$-$(v$\times$w)

so w$\times$v=$-$(5i+5j+5k) =-5i-5j-5k

Step 4. w×w and v×v

According to the algebraic properties of cross product the cross product of any vector with itself is the zero vector

(a ×a=0).

Hence w ×w=0 and v ×v=0