#### 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