Question
Mon April 16, 2012 By: Avi Wadhwa

please give me a mathematical proof of the following expression- vector 'a' cross vector 'b' is equal to modulus of vector 'a' dot modulus of vector 'b' dot 'sin c' (i wrote down the question in words rather than in the form of mathematical expression..) where 'c' is the angle between vector 'a' and vector 'b'..?

Expert Reply
Tue April 17, 2012

The cross product a × b is defined as a vector c that is perpendicular to both a and b, with a direction given by the right-hand rule and a magnitude equal to the area of theparallelogram that the vectors span.

The cross product is defined by the formula

where ? is the measure of the smaller angle between a and b (0° ? ? ? 180°), |a| and |b| are the magnitudes of vectors a and b, and n is a unit vector perpendicular to the plane containing a and b in the direction given by the right-hand rule as illustrated. If the vectors a and b are parallel (i.e., the angle ? between them is either 0° or 180°), by the above formula, the cross product of a and b is the zero vector 0.

The direction of the vector n is given by the right-hand rule, where one simply points the forefinger of the right hand in the direction of a and the middle finger in the direction of b. Then, the vector n is coming out of the thumb (see the picture on the right). Using this rule implies that the cross-product is anti-commutative, i.e., b × a = ?(a × b). By pointing the forefinger toward b first, and then pointing the middle finger toward a, the thumb will be forced in the opposite direction, reversing the sign of the product vector.

Using the cross product requires the handedness of the coordinate system to be taken into account (as explicit in the definition above). If a left-handed coordinate system is used, the direction of the vector n is given by the left-hand rule and points in the opposite direction.

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