# Help with Calculating the Cross and Dot Product

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• Sep 12th 2012, 11:00 AM
biglew404
Help with Calculating the Cross and Dot Product
I have figured out how to determine the Dot product becasue it is fairly straight forward. However the Cross product seems to be a challenge for me as I cannot seem to grasp.

Here is the Problem

Determine: AXB if A = 3i - 2j and B = i + 2k

Need help working out this problem......
• Sep 12th 2012, 11:21 AM
kalyanram
Re: Help with Calculating the Cross and Dot Product
Consider $\displaystyle A = u+v, B=x+y$ where $\displaystyle u=3i, v=-2j, x=i \hspace{2mm}and \hspace{2mm}y=2k$.
Spoiler:

Now $\displaystyle A$x$\displaystyle B$ = $\displaystyle (u+v)$x$\displaystyle (x+y)$ = $\displaystyle u$x$\displaystyle (x+y)+v$x$\displaystyle (x+y)$(why?)
$\displaystyle \implies u$x$\displaystyle x$+$\displaystyle u$x$\displaystyle y$ + $\displaystyle v$x$\displaystyle x$+$\displaystyle v$x$\displaystyle y$(why?)
Now substitute their respective values and simplify.
• Sep 12th 2012, 11:36 AM
Plato
Re: Help with Calculating the Cross and Dot Product
Quote:

Originally Posted by biglew404
I have figured out how to determine the Dot product becasue it is fairly straight forward. However the Cross product seems to be a challenge for me as I cannot seem to grasp.
Here is the Problem
Determine: AXB if A = 3i - 2j and B = i + 2k

Evaluate the determinate by expanding alone the first row:
$\displaystyle A\times B=\left| {\begin{array}{*{20}{c}} i&j&k \\ 3&{ - 2}&0 \\ 1&0&2 \end{array}} \right|=~?$
• Sep 12th 2012, 01:44 PM
HallsofIvy
Re: Help with Calculating the Cross and Dot Product
That is equivalent to the basic definition of x=cross product:
$\displaystyle i\times j= k$, $\displaystyle j\times k= i$, $\displaystyle k\times i= j$.
cross product distributes over addition, and cross product is skew- symmetric ($\displaystyle u\times v= -v\times u$).

So $\displaystyle (ai+ bj+ ck)\times (pi+ qj+ rk)= (ai)\times(pi+ qj+ rk)+ (bj)\times(pi+ qj+ rk)+ (ck)\times (pi+ qj+ rk)$
$\displaystyle = ap (i\times i)+ aq (i\times j)+ ar(i\times k)+ bp(j\times i)+ bq(j\times j)+ br(j\times k)+ cp(k\times i)+ cq(k\times j)+ cr(k\times k)$
Because cross product is skew symmetric, $\displaystyle i\times i= 0$, $\displaystyle j\times j= 0$, $\displaystyle k\times k= 0$, $\displaystyle i\times k= -k\times i= -j$, $\displaystyle j\times i= -i\times j= -k$, and $\displaystyle k\times j= -j\times k= -i$.

That reduces the above to [tex](aq- bp)i- (ar- cp)j+ (aq- bp)k which is the same as
$\displaystyle \left|\begin{array}{ccc}i & j & k \\a & b & c \\ p & q & r\end{array}\right|$
• Sep 12th 2012, 01:45 PM
biglew404
Re: Help with Calculating the Cross and Dot Product
(-4 - 0)i - (6 - 0)j + (0 - -2)k
= -4i - 6j + 2K

Thanks Alot..........I understand better now