Cross Or Vector Product Of Vectors


Cross product of any two vectors P and § is vector R. Cross product of vectors takes the following dimensions of analysis.

(i) The magnitude of R is given by R = PQ sin Θ

Θ being the acute angle between P and Q(i.e. sin q is always a positive quantity).

(ii) Its direction is perpendicular to the plane of P and Q.

(iii) Its sense is given by the direction of advance of a right hand screw which is rotated from the direction of first vector to the second vector. If n is the unit vector in the direction of R,

   Vector Product Of Vectors.JPG

Then we can write

P × Q = PQ sin q n


From the definition of cross product, it is obvious that

P × Q = –Q × P

magnitude remains.JPG

Since in both cases the magnitude remains the same the direction reverses.


Cross product of two vectors is a vector.


magnitude remains1.JPG

Figure clearly shows that the magnitude of the cross product

            = PQ sin Θ

            = P(Q sin Θ)


            = Area of parallelogram with P and Q as sides.

Noting that i, j and k are projections of unit vectors in three mutually perpendicular directions taken according to the right handed coordinate system and, by definition of the cross product,

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

       j × i = –k                       k × j = –i                       and       i × k = –j

The cross product of a collinear vector must vanish since in such a case q = 0.

∴       i × i = j × j = k × k = 0


     P × Q = (Pxi + Pyj + Pzk) × (Qxi + Qyj + Qzk)

              = Px Qy k + Px Qz (–j) + Py Qx (–k) + Py Qz i + Pz Qz i + Pz Qx j + Pz Qy (–i)

              = (Py Qz – Pz Qy) i + (Pz Qx – Px Qz)j + (Px Qy – Py Qx)k

cross product of two vectors.JPG

The cross product of two vectors may also be used to find the angle between the two vectors. Here,

sin Θ = |P × Q| / PQ

where |P × Q| is the magnitude of P × Q.

We have seen that the commutative law does not hold good for the cross product.

P × Q  ¹ Q × P

But, the distributive law holds good here:

P × (Q × R) = P × Q + P × R

and       n(P × Q) = nP × Q = P × nQ


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