# [SOLVED] show zero x vector in v3, (cross product)

• February 23rd 2009, 09:01 PM
rcmango
[SOLVED] show zero x vector in v3, (cross product)
must show that 0 x a and a x 0 = 0 for any vector a in V3

so would this be sufficient:

<0, 0, 0> x <1, 2, 3> = 0

and <2, 3, 4> x <0, 0, 0> = 0

because

0 - 0 + 0 = 0

correct?

believe this is cross product stuff.
• February 23rd 2009, 09:11 PM
o_O
No. All you have shown is that this is true for a few vectors. You need to prove that it is true for all vectors.

To show this, you let $a$ be any general vector with components: $a = \left< a_1, a_2, a_3\right>$

So let's take the cross product:
$0 \times a = \left| \begin{matrix} i & j & k \\ 0 & 0 & 0 \\ a_1 & a_2 & a_3 \end{matrix} \right| = \left| \begin{matrix} 0 & 0 \\ a_2 & a_3 \end{matrix} \right|i - \left| \begin{matrix} 0 & 0 \\ a_1 & a_3 \end{matrix} \right|j + \left| \begin{matrix} 0 & 0 \\ a_1 & a_2 \end{matrix} \right|j = \cdots$

Do the same with $a \times 0$.

Both of these should be equal to 0 and since $a$ was a general vector, then it must be true for any vector in $\mathbb{R}^3$.
• February 23rd 2009, 10:40 PM
rcmango
thanks, the other would be
i, j, k
|a1, a2, a3|
|0, 0, 0|

and the result 0 for both.