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theperthvan
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Why is it possible to take the cross product in only 3 and 7 dimensions?
prasannapakkiam said:Well, the idea is that a matrix is created. The determinent can do very funny things. Just try to find the determinents of 3x3, 4x4, 5x5, 6x6, 7x7. You may figure out why...
A cross product is a mathematical operation that takes two vectors as input and produces a third vector that is perpendicular to both of the input vectors. It is commonly used in 3D and 7D geometry to calculate the direction of a surface normal or to find a vector that is perpendicular to two given vectors.
To calculate a cross product in 3D, you can use the formula:
A x B = (AyBz - AzBy, AzBx - AxBz, AxBy - AyBx)
Where A and B are the two input vectors and x represents the cross product operator. This formula is also known as the "right-hand rule" and can be used to determine the direction of the resulting vector.
Yes, a cross product can be calculated in 7D using a similar formula as in 3D. The only difference is that instead of three components, the resulting vector will have seven components. The formula can be generalized as:
A x B = (A2B3 - A3B2, A3B1 - A1B3, A1B2 - A2B1, A4B5 - A5B4, A5B6 - A6B5, A6B7 - A7B6, A7B1 - A1B7)
Where A and B are the two input vectors and x represents the cross product operator.
Yes, calculating cross products is used in many fields such as physics, engineering, and computer graphics. In physics, cross products are used to calculate the torque and angular momentum of an object. In engineering, they are used to determine the force exerted on an object by a magnetic field. In computer graphics, they are used to create 3D models and animations.
Yes, a cross product results in a vector while a dot product results in a scalar. In other words, a cross product gives us information about the direction of two vectors while a dot product gives us information about their magnitude. Additionally, a dot product is commutative (A · B = B · A) while a cross product is anti-commutative (A x B = -B x A).