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brinlin
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Scalar Triple Product and Coplanarity
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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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Evgeny.Makarov
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Welcome to the forum. Before we go into the discussion of your particular question, please read the https://mathhelpboards.com/help/forum_rules/, especially "Show the nature of your question in your thread title" and "Show some effort". In this case, you should probably explain what exactly is not clear to you.
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skeeter
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HOI
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Do you not know how to do a cross product and a dot product?
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 $u= \vec{i}+ 2\vec{j}+ 3\vec{k}$.
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 $u= \vec{i}+ 2\vec{j}+ 3\vec{k}$.
1. What is the definition of scalar triple product?
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.
2. How is scalar triple product used in physics?
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.
3. What is the geometric interpretation of scalar triple product?
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.
4. What is the relationship between scalar triple product and coplanarity?
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.
5. What are some real-life applications of scalar triple product and coplanarity?
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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