# Product of Two Vectors - Vector (Or Cross) Product of Two Vectors

#### definition

The vector product of two nonzero vectors , is denoted by and defined as  vec a xx vec b = |vec a|  |vec b| sin theta   hat n,
where ,θ is the angle between , vec a and vec b = 0 ≤ θ ≤ π and hat n is a unit vector perpendicular to both vec a and vec b, such that vec a, vec b and hat n form a right handed system in following fig .

i.e., the right handed system rotated from vec a to vec b moves in the direction hat n.
If either vec a = vec 0 or vec b = vec 0 , then θ is not defined and in this case, we define vec a xx vec b = vec 0.

#### notes

Observations:
1) vec a xx vec b is a vector.

2) Let vec a and vec b be two  nonzero vectors. Then vec a xx vec b = vec 0 if and only if vec a and vec b are parallel (or collinear) to each other, i.e.,
vec a xx vec b = vec 0 <=>  vec a||vec b
In particular , vec a xx vec b = vec 0 and  vec a xx (-vec a) = vec 0, since in the first situation, θ = 0 and in the second one, θ = π, making the value of sinθ to be 0.

3) If  , θ = π/2 then vec a xx vec b = |vec a||vec b|.

4)  In view of the Observations 2 and 3, for mutually perpendicular unit vectors hat i , hat j and hatk fig.

hat i xx hat i = hat j xx hat j = hat k xx hat k = vec 0
hat i xx hat j = hat k ,  hat j xx hat k = hat i , hat k xx hat i = hat j

5) In terms of vector product, the angle between two vectors vec a and vec b may be given as
sin theta = (|vec a xx vec b|)/(|vec a||vec b|)

6)  It is always true that the vector product is not commutative, as vec a xx vec b = - vec b xx vec a.
Indeed vec a xx vec b = |vec a||vec b|  sin theta   hat n_1, where vec b, vec a   "and"   hat n_1 form a right handed system, i.e., θ is traversed from vec b   "to"  vec a  in following fig.

While , vec b xx vec a = |vec a||vec b|   sin theta   hat n_1 , where vec b, vec a  "and"   hat n_1 form a right handed system i.e. θ is traversed from vec b " to"   vec a,
Fig.

Thus, if we assume vec a and vec b to lie  in the plane of the paper, then hat n and hat n_1 both will be perpendicular to the plane of the paper. But, hat n  being directed above the paper while hat n_1 directed below the paper i.e. hat n_1 = -hat n.
Hence vec a xx vec b = |vec a||vec b|  sin theta  hat n
= - |vec a||vec b|  = sin theta  hatn_1
= -vec b xx vec a =

7) In view of the Observations 4 and 6, we have
hat j xx hat i = - hat k , hat k xx hat j = - hat i and  hat i xx hat k = -hat j.

8) If vec a and vec b represent the adjacent sides of a triangle then its area is given as 1/2 |vec a xx vec b|.
By definition of the area of a triangle, we have from fig.

Area of triangle ABC =  1/2 AB .CD
But AB = |vec b| (as given ), and  CD = |vec a|   sin θ.

Thus,  Area of triangle ABC =  1/2 |vec b||vec a| sin theta = 1/2 | vec a xx vec b|.

9)  If vec a  "and"   vec b represent the adjacent sides of a  parallelogram, then its area is given by |vec a xx vec b|.From the following Fig. we have

Area of parallelogram ABCD = AB. DE.
But AB = |vec b| (as given), and
DE = |vec a|   sin theta
Thus,  Area of parallelogram ABCD = |vec b| |vec a|  sin theta = | vec a xx vec b|.
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CROSS PRODUCT [00:34:02]
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