The cross product or vector product of two vectors a ⨯ b is a vector that is perpendicular to both a and b whose direction is given by the right-hand rule:
As shown in the previous figure, in order to apply the right-hand rule one simply points the palm of the right hand in the direction of a and closes it in the direction of b. The direction of the cross product of both vectors is coming out of the thumb.
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The magnitude of the cross product is given by:
From the previous expression it can be deduced that the cross product of two parallel vectors is 0.
The cross product is anti-commutative; if we apply the right-hand rule to multiply b ⨯ a it gives:
This vector has the same magnitude as a ⨯ b, but points in the opposite direction. And two vectors are equal only if they have both the same magnitude and direction.
From the previous results we can determine the cross products of the standard unit vectors i, j, k:
When two vectors are given in unit vector notation , their cross product is given by the following determinant:
And expanding the previous expression:
The magnitude of the cross product is equal to the area of the parallelogram that the vectors span. Viewing both from above:
The area of a parallelogram is its base multiplied by its height, therefore:
In the following animation, you can move both vectors a and b in the xy-plane to see their cross product.
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