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brinlin

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In summary, the conversation discusses how to calculate the cross product and dot product of two vectors, with an example of v= <2, 3, 1> and w= <3, 1, 2>. The cross product is calculated using the determinant formula, while the dot product is calculated by taking the dot product of the cross product with another vector, u= <1, 2, 3>.

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brinlin

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Evgeny.Makarov

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With v= <2, 3, 1> and w= <3, 1, 2> the cross product, v x w, can be calculated as the determinant

$\left|\begin{array}{ccc}\vec{i} & \vec{j} & \vec{k} \\ 2 & 3 & 1 \\ 3 & 1 & 2 \end{array}\right|= \vec{i}\left|\begin{array}{cc}3 & 1 \\ 1 & 2\end{array}\right|- \vec{j}\left|\begin{array}{cc}2 & 1 \\ 3 & 2\end{array}\right|+ \vec{k}\left|\begin{array}{cc} 2 & 3 \\ 3 & 1 \end{array}\right|$

$= (6- 1)\vec{i}- (4- 3)\vec{j}+ (2- 9)\vec{k}= 5\vec{i}- \vec{j}- 7\vec{k}$

Now take the dot product of that with $u= \vec{i}+ 2\vec{j}+ 3\vec{k}$.

The scalar triple product is a mathematical operation that involves three vectors and results in a scalar value. It is calculated by taking the dot product of one vector with the cross product of the other two vectors.

In physics, scalar triple product is used to calculate the volume of a parallelepiped formed by three vectors. It is also used in determining the moment of inertia of a rigid body and in solving problems related to torque and angular momentum.

The scalar triple product represents the volume of a parallelepiped formed by three vectors in three-dimensional space. It can also be interpreted as the signed volume of a tetrahedron with one vertex at the origin and the other three vertices at the end points of the three vectors.

If the scalar triple product of three vectors is equal to zero, then the three vectors are coplanar, meaning they lie in the same plane. This is because the volume of a parallelepiped formed by coplanar vectors is zero.

Scalar triple product and coplanarity have various applications in fields such as engineering, physics, and computer graphics. They are used in calculating the moment of inertia of objects, determining the stability of structures, and in computer graphics for 3D modeling and animation.

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