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physics kiddy
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Why is dot product given by a*b cosθ whereas cross product ab sinθ.
The cross product and dot product are both mathematical operations used in vector algebra. The main difference between them is that the cross product results in a vector, while the dot product results in a scalar. This means that the cross product takes two vectors and produces a third vector that is perpendicular to both, while the dot product takes two vectors and produces a scalar value that represents the magnitude of the projection of one vector onto the other.
The cross product is useful for determining the direction of a vector perpendicular to two other vectors, which is important in many physical applications such as calculating torque or magnetic fields. The dot product, on the other hand, is useful for calculating the angle between two vectors or determining the component of one vector in the direction of another. It is also used in many mathematical and engineering calculations.
The cross product is calculated by taking the determinant of a 3x3 matrix composed of the two vectors and their unit vectors. The dot product is calculated by multiplying the corresponding components of the two vectors and then adding them together. Both operations have specific formulas and can also be calculated using geometric methods.
Yes, there are several properties of these operations that are important to know. The cross product is anti-commutative, meaning that changing the order of the vectors will result in a vector in the opposite direction. It is also distributive and follows the right-hand rule. The dot product is commutative, meaning that changing the order of the vectors does not change the result. It is also distributive and follows the law of cosines.
Yes, both the cross product and dot product can be used in higher dimensions, but they are not as commonly used as in three dimensions. In higher dimensions, the cross product involves taking the determinants of larger matrices and the dot product involves multiplying more components. In some cases, alternative operations may be used instead, such as the wedge product for the cross product in four dimensions.