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Jhenrique
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If u × v = u × w, so v = w ?
Jhenrique said:If u × v = u × w, so v = w ?
Not necessarily. What if u is the zero vector?mathman said:u × v = u × w is equivalent to u × (v-w) =0.
Therefore u is parallel to v-w, so that v-w is a multiple of u.
D H said:Not necessarily. What if u is the zero vector?
The cross product is a mathematical operation that takes two vectors as input and produces a third vector that is perpendicular to both input vectors. When we say that the cross product is cancellative, we mean that if the cross product of two vectors is equal to the cross product of two other vectors, then the original two vectors must also be equal.
The cancellative property of the cross product allows us to simplify equations involving vectors and makes it easier to solve problems in physics and engineering. It also helps us to identify equivalent vectors and simplify geometric proofs.
Yes, there are some exceptions to this property. It only holds true for three-dimensional vectors and does not apply to vectors in higher dimensions. Additionally, the property may not hold if the vectors are parallel or collinear.
The cancellative property can be proven using the properties of determinants and the properties of vector multiplication. By expanding the determinants of the cross products, we can show that they are equal only if the original vectors are also equal.
The cross product is cancellative property has various real-world applications in physics, engineering, and computer graphics. It is used in calculating the torque exerted by a force on a rigid body, determining the direction of magnetic fields, and in 3D computer graphics to calculate lighting and shading effects on objects.