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Noxide
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Why is the cross product defined this way
(blah)i + (blah)j + (blah)k = u
and used this way
(blah)i - (blah)j + (blah)k = u
(blah)i + (blah)j + (blah)k = u
and used this way
(blah)i - (blah)j + (blah)k = u
HallsofIvy said:Making the second coeffient "-" makes it fit a nice mnemonic:
expanding a determinant along the top row
[tex]\left|\begin{array}{ccc}a & b & c \\ d & e & f\\ g & h & i\end{array}\right|= a\left|\begin{array}{cc}e & f \\ h & i\end{array}\right|- b\left|\begin{array}{cc}d & f \\ g & i\end{array}\right|+ c\left|\begin{array}{cc}d& e \\ g & h\end{array}\right|[/tex]
If [itex]\vec{u}= a\vec{i}+ b\vec{j}+ c\vec{k}[/itex] and [itex]\vec{v}= x\vec{i}+ y\vec{j}+ z\vec{k}[/itex], then writing [itex]\ve{u}\times \vec{v}= (bz-cy)\vec{i}+ (az- cx)\vec{j}+ (ay- bx)\vec{k}[/itex] as [itex](bz-cy)\vec{i}- (cx- az)\vec{j}+ (au= bx-)\vec{k}[/itex] makes it clearer that we can think of it as
[tex]\vec{u}\times\vec{v}= \left|\begin{array}{ccc}\vec{i} & \vec{j} & \vec{k} \\ a & b & c \\ x & y & z\end{array}\right|[/tex]
The cross product is a mathematical operation that takes two vectors as input and produces a third vector that is perpendicular to both of the input vectors. It is denoted by the symbol "x" and is also known as the vector product.
The cross product is defined as the product of the magnitudes of the two vectors and the sine of the angle between them. This can be represented as: A x B = |A| |B| sin(theta), where A and B are the two input vectors and theta is the angle between them.
The cross product has a geometric interpretation in terms of the right-hand rule. If the index finger of the right hand is pointed in the direction of the first vector and the middle finger is pointed in the direction of the second vector, then the thumb will point in the direction of the resulting cross product vector.
The cross product is used in physics and engineering to calculate various quantities, such as torque and angular momentum. It is also used in the calculation of magnetic fields and in the motion of charged particles in electromagnetic fields.
The cross product has several important properties, including being anti-commutative (A x B = -B x A), distributive (A x (B + C) = A x B + A x C), and being perpendicular to both of the input vectors. It is also equal to zero if the two vectors are parallel or anti-parallel.