notReallyHere said:Homework Statement
[tex] \bigtriangledown\times\\(v\times w)= v(\bigtriangledown\cdot w) - w(\bigtriangledown\cdot v)+ (v\cdot\bigtriangledown)w - (w\cdot\bigtriangledown) v [/tex]
I've tried expanding left side and get
[v1(dw2/dy+dw3/dz)-w1(dv2/dy+dv3/dz)]i +
[v2(dw3/dz+dw1/dx)-w2(dv3/dz+dv1/dx)]j +
[v3(dw1/dx+dw2/dy)-w3(dv2/dy+dv1/dx)]k
is this right way to start it? another way to attack it?
The vector cross product identity, also known as the triple vector product identity, is a mathematical equation that relates the cross product of three vectors to the dot product of those vectors. It states that the cross product of the cross product of two vectors is equal to the difference between the dot product of those vectors and the dot product of the first vector with the cross product of the second vector.
The vector cross product identity is important because it allows us to simplify complex vector equations and manipulate them in a more efficient manner. It is also a fundamental concept in vector calculus and is used in many areas of physics, engineering, and computer graphics.
The vector cross product identity can be derived using the properties of vector operations such as the distributive, commutative, and associative properties. It can also be derived using the geometric properties of the cross product and the properties of the dot product. The proof involves expanding both sides of the equation and simplifying them to show that they are equal.
Yes, the vector cross product identity can be extended to higher dimensions. In three-dimensional space, the cross product results in a vector, but in higher dimensions, it results in a higher-dimensional object such as a bivector or a trivector. The concept of the vector cross product identity can also be extended to these higher-dimensional objects.
The vector cross product identity has many applications, including in physics for calculating torque and angular momentum, in engineering for calculating moments of inertia and angular velocity, and in computer graphics for calculating the direction and intensity of light in 3D scenes. It is also used in vector calculus to solve problems involving vector fields and line integrals.