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Difference between Dot product and vector product ?
Answers
Best Explanation I can give.
The dot product is a scalar. The dot product of two vectors gives you the value of the magnitude of one vector multiplied by the magnitude of the projection of the other vector on the first vector.
If a⃗ and b⃗ are two vectors with an angle θ between them, then the dot product of a⃗ and b⃗ is a⃗ .b⃗ =|a⃗ ||b⃗ |cosθ .
This can be considered as |a⃗ |(|b⃗ |cosθ) , where |a⃗ | is the magnitude of a⃗ and (|b⃗ |cosθ) is the magnitude of the projection of b⃗ on a⃗ .
This can also be considered as | b⃗ | (| a⃗ | \cos \theta), where | b⃗ | is the magnitude of b⃗ and (| a⃗ | \cos \theta) is the magnitude of the projection of a⃗ on b⃗ .
The cross product is a vector. The magnitude of the cross product of two vectors is the magnitude of one vector multiplied by the magnitude of the projection of the other vector in the direction orthogonal to the first vector.
If a⃗ and b⃗ are two vectors with an angle θ between them, then the cross product of a⃗ and b⃗ is a⃗ × b⃗ = | a⃗ | | b⃗ | \sin \theta.
The direction of the resultant of the cross product of two vectors perpendicular to the plane of the two vectors whose cross product is being taken considering the right hand rule i.e. if you keep your thumb perpendicular to the other fingers and move the other fingers from the first vector to the second vector the direction of the resultant vector is the direction of the thumb.
The geometric interpretation of the magnitude of the cross product of two vectors a⃗ and b⃗ is that it is equal to the area of the parallelogram formed by the vectors a⃗ and b⃗ .
It does not make any sense to say that “the dot product’s answer is limited to two dimensions” since the result of the dot product of two vectors is a scalar which does not have any direction.
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