Question

find (a) v × w, (b) w × v, (c) w × w, and

(d) v × v

v = 3i + j + 3k

w = i - k

Short answer

v ×w= -i +6j -k

w ×v= i -6j +k

w ×w=0

v ×v=0

Step-by-step solution

Step 1. Given information

v = 3i + j + 3k

w = i - k

Step 2. v× w  

For two vectors

v=a₁i+b₁j+c₁k

w=a₂i+b₂j+c₂k

cross product v× 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 = 3i + j + 3k

w = i - k

$$\begin{gathered} v\times w=\left|\begin{array}{ccc}i& j& k\\ 3& 1& 3\\ 1& 0& -1\end{array}\right| \\ =i\left|\begin{array}{cc}1& 3\\ 0& -1\end{array}\right|-j\left|\begin{array}{cc}3& 3\\ 1& -1\end{array}\right|+k\left|\begin{array}{cc}3& 1\\ 1& 0\end{array}\right| \\ =-i+6j-k \end{gathered}$$

Step 3. w × v 

According to the algebraic properties of cross product

w × v = -(v × w)

so w × v=$-$(-i+6j-k)= i -6j +k

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