Share
Notifications

View all notifications
Advertisement

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

Login
Create free account


      Forgot password?

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|`. 

Video Tutorials

We have provided more than 1 series of video tutorials for some topics to help you get a better understanding of the topic.

Series 1


Series 2


Shaalaa.com | CROSS PRODUCT

Shaalaa.com


Next video


Shaalaa.com


CROSS PRODUCT [00:34:02]
S
Series 1: playing of 2
1
0%


Advertisement
View in app×