please explain me obout cross product of vectos n detail

Dear aditya,

The cross product of two vectors a and b is denoted by a × b. In physics, sometimes the notation a∧b is used,   though this is avoided in mathematics to avoid confusion with the exterior product.

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 the parallelogram that the vectors span.

The cross product is defined by the formula

where θ is the measure of the smaller angle between a and b (0° ≤ θ ≤ 180°), a and b are the magnitudes of vectors a and b, and 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 θ 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 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 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.

i × j = k           j × k = i           k × i = j

j × i = −k           k × j = −i           i × k = −j

i × i = j × j = k × k = 0.

The cross product can be calculated by distributive cross-multiplication:

a × b = (a₁i + a₂j + a₃k) × (b₁i + b₂j + b₃k)

a × b = a₁i × (b₁i + b₂j + b₃k) + a₂j × (b₁i + b₂j + b₃k) + a₃k × (b₁i + b₂j + b₃k)

a × b = (a₁i × b₁i) + (a₁i × b₂j) + (a₁i × b₃k) + (a₂j × b₁i) + (a₂j × b₂j) + (a₂j × b₃k) + (a₃k × b₁i) + (a₃k × b₂j) + (a₃k × b₃k).

GOODLUCK

Please feel free to post as many doubts on our discussion forum as you can. We are all IITians and here to help you in your IIT JEE preparation.

Win exciting gifts by answering the questions on Discussion Forum..

Approved