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Scorching goIITian

Joined:	19 Feb 2010
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24 Feb 2010 14:38:40 IST

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why the resultant of the cross product directs perpendicularly in or out of the plane ????.

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why the resultant of the cross product directs perpendicularly in or out of the plane ????.

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edison

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Joined: 19 Oct 2006

Posts: 7537

24 Feb 2010 15:35:28 IST

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Because vector multiplication in mathematics is defined in such a manner that the resultant vector is perpendicular the the plane containing the two vectors of which vector product is sought.

Suppose we have a = a₁ i + a₂ j + a₃ k and b = b₁ i + b₂ j + b₃ k.

Then a x b = (a₁ i + a₂ j + a₃ k) x (b₁ i + b₂ j + b₃ k).

It can be shown that it is all right to work this out by working out all the 9 separate little cross products that we get from multiplying these two brackets together. Doing this, and using the results above, we get

a x b = (a₂b₃ - a₃b₂) i + (a₃b₁ - a₁b₃) j + (a₁b₂ - a₂b₁) k

and this gives us the rule for how we work out a cross product using components.

Here is a numerical example using this result.
A force F = 3i + 2j + 4k acts through the point with position vector r = 2i + j + 3k.
What is its torque about a perpendicular axis through O?

The torque = r x F = (1x4 - 3x2) i + (3x3 - 2x4) j + (2x2 - 1x3) k = - 2i + j + k.

these unit vectors and the right-handrule

Since sin 0 = 0 and sin 90 = 1 and each vector is of unit length, we have
i x i = j x j = k x k = O, (the zero vector).
Also, i x j = k and j x k = i and k x i = j

while j x i = - k and k x j = - i and i x k = - j.
You can see how the various plus and minus signs come by using the right-hand rule for each product.

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