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Mathematics
OpenStudy (anonymous):

Hello What is the difference between a dot product and a cross product and how do i find a vector parallel to (2,-3,6)

OpenStudy (anonymous):

Dot products and cross products are very different. Look at the result: one gives a scalar, the other gives another vector.

OpenStudy (turingtest):

...and multiplying a vector by any positive scalar will not change it's direction

OpenStudy (anonymous):

The dot product is computed as follows: \[ \left[\begin{matrix} a_1\\a_2\\a_3 \end{matrix}\right]\cdot\left[\begin{matrix} b_1\\b_2\\b_3 \end{matrix}\right]=a_1b_1+a_2b_2+a_3b_3 \]The pattern remains the same for larger vectors.

OpenStudy (anonymous):

\[a= \left(\begin{matrix}a_1 \\ a_2 \\ a_3\end{matrix}\right) \text{ , } b =\left(\begin{matrix}b_1 \\ b_2 \\ b_3\end{matrix}\right)\] \[a . b = |a||b|\cos \theta = a_1b_1 + a_2b_2 + a_3b_3\]

OpenStudy (anonymous):

The cross product can be computed as the determinant of the following matrix:\[ \left|\begin{matrix} i&j&k\\a_1&a_2&a_3\\b_1&b_2&b_3 \end{matrix}\right| \]

OpenStudy (anonymous):

\[a \times b = \det\left[\begin{matrix}i & j & k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3\end{matrix}\right]\]

OpenStudy (anonymous):

damnit you're too quick! XD

OpenStudy (anonymous):

Hahaha XD

OpenStudy (turingtest):

ok I'm slow with latex...

OpenStudy (anonymous):

haha

OpenStudy (turingtest):

\[\left|\begin{matrix} \hat i&\hat j&\hat k\\a_1&a_2&a_3\\b_1&b_2&b_3 \end{matrix}\right|\]I added the hats :)

OpenStudy (anonymous):

Nice. Alternatively, boldface!\[ \left|\begin{matrix} \mathbf{i}&\mathbf{j}&\mathbf{k}\\a_1&a_2&a_3\\b_1&b_2&b_3 \end{matrix}\right| \]

OpenStudy (turingtest):

\mathbf news to me, thanks

OpenStudy (anonymous):

thanks bouscal and turning and eigen

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