What is the cross product of a vector with itself?

Question

What is the cross product of a vector with itself?

What is the cross product of a vector with its opposite?

Answer 1

In both cases zero. The magnitude of the cross product is given by the product of the magnitudes of the two vectors times the sine of the angle between them. In the event that the two vectors are equal, the angle between them is 0, and where the two vectors are opposite, the angle between them is π. In either case, the sine of the angle is 0, so the magnitude of the cross product is 0, making it the zero vector.

Answer 2

Cross Product Of Parallel Vectors

Answer 3

In both cases, you will get zero. As you may be aware, when you take the dot product of two perpendicular vectors you get zero. For the cross product, you have the opposite case. The cross product of parallel vectors is zero.

These properties are due to the fact that the dot product cas be described with cosine (which is zero at 90 degrees) and the cross product can be described by sine (which is zero at 0 and 180 degrees).

Answer 4

Zero.

The magnitude of the cross product is given by:

|| a X b || = || a || || b || sinθ

where θ is the angle between them. The angle between a vector and itself is zero. So the cross product of a vector with itself is zero.

|| a X a || = || a ||² sin(0) = || a ||² * 0 = 0
_________________

The cross product of a vector with it opposite is also zero. We would have two vectors a and -a. The angle between them is π. The magnitude of their cross product is:

|| a X -a || = || a || || -a || sin(π) = || a ||² * 0 = 0

Answer 5

let the vector be A. I---------------->
then you know cross product of A is
AcrossA = IAI IAI sin(a), where a is the angle between the vectors. in this case the angle is 0 and I I represents modulus,ie, positive value or magnitude.
we know, sin0 = 0. hence, the result is 0.

if the 2nd vector be opposite, then it can be expressed as -A vector. <------------I------------> and the angle between A and -A is 180 degree.
so taking the cross product, we find it is IAI IAI sin 180 ,ie, IAI IAI sin (180-0) ,ie, IAI IAI sin 0, ie, 0.

for your info, a cross product can have a definite vector value only when angle a has a value. a vector has only a specific direction in a cartesian system. there cannot be two equal vectors inclined to each other at a certain angle. this will not be possible as they will coincide and represent the same vector. so a is 0 and so also the cross product.

Answer 6

0

Answer 7

Zero, in each case. Two vectors can have a non-zero cross product only if they are not collinear.

Answer 8

No such thing Cross PRODUCT as in multiplication You can't just say "Cross product of A" unless by "A by itself" u meant "A by A" in which case: A x A = 0 If A x B = C, C is perpendicular to both A and B

Answer 9

cross product of vector with itself is
a^2 if the vector is a
as axa =aacosine of angle between them
as the angle is 0
cos0=1

Answer 10

zero (since a cross a=a*a sin x)


Using property...a cross (-b)=-(a cross b),
second ans is also "zero"