Scalar triple product coplanarity

In summary: I would use "scalar triple product" instead of "u·(v×w)" but that's just a minor detail.)In summary, to show that u, v, and w lie in the same plane in R3, it must be proven that if u · (v × w) = 0, then u is orthogonal to vxw and vxw is orthogonal to v and w. This means that u must lie in the same plane as v and w, making them coplanar. This can be proven mathematically using the concept of the scalar triple product.
  • #1
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Homework Statement



Show that u, v, w lie in the same plane in R3 if and only if u · (v × w) = 0.


Homework Equations





The Attempt at a Solution



if u · (v × w) = 0, then u is orthogonal to vxw, and
vxw is orthogonal to v and w.

therefore, u must lie in the same plane determined by v and w (i.e. they are coplanar)


Is this correct? Also, how would i describe this proof mathematically?

Thanks!
 
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  • #2
u · (v × w) = 0

use geometrical proof

translate the equation as volume of parallopiped
the volume is zero only when the height is zero
or all vectors lie in the same plane
 
  • #3
Neen87 said:

Homework Statement



Show that u, v, w lie in the same plane in R3 if and only if u · (v × w) = 0.


Homework Equations





The Attempt at a Solution



if u · (v × w) = 0, then u is orthogonal to vxw, and
vxw is orthogonal to v and w.

therefore, u must lie in the same plane determined by v and w (i.e. they are coplanar)


Is this correct? Also, how would i describe this proof mathematically?

Thanks!
Yes, that's correct and, once you have stated exactly HOW you know that two vectors, perpendicular to the same non-zero vector, are co-planar, it IS "mathematical".

payumooli's suggestion is another way to do it but since you have already done it your own way, I suggest you stay with it.
 

1. What is the definition of scalar triple product coplanarity?

The scalar triple product coplanarity is a mathematical concept used in vector calculus to determine whether three vectors in three-dimensional space lie on the same plane or not. It is also known as the triple scalar product or the box product.

2. How is the scalar triple product calculated?

The scalar triple product is calculated by taking the dot product of one vector with the cross product of the other two vectors. This can be represented by the formula (a ⃗ ⋅ (b ⃗ x c ⃗)), where a, b, and c are the three vectors involved.

3. What does it mean when the scalar triple product is equal to 0?

If the scalar triple product is equal to 0, it means that the three vectors are coplanar, or in other words, they lie on the same plane. This is because the dot product of a vector with a perpendicular vector is always 0, and the cross product of two parallel vectors is also 0, resulting in a scalar triple product of 0.

4. How is scalar triple product coplanarity used in physics?

In physics, the scalar triple product coplanarity is used to determine if three forces acting on an object are balanced or not. If the scalar triple product of the three forces is equal to 0, it means that the forces are coplanar and the object will be in equilibrium. If the scalar triple product is not equal to 0, it means that there is a net torque acting on the object, causing it to rotate.

5. Can the scalar triple product be used for more than three vectors?

No, the scalar triple product is only defined for three vectors in three-dimensional space. However, the concept of coplanarity can be extended to more than three vectors by using the generalization known as the scalar n-tuple product, which is defined for n vectors in n-dimensional space.

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