Why is the scalar triple product of coplanar vectors zero?

Why is the scalar triple product of coplanar vectors zero?

If three vectors are coplanar thhen they must lie in the same plane… There is no such vector which has Depth or height… i.e , Height = 0… Thus Scalar trople product is zero if thre vectors are coplanar…

How do you prove the triple scalar product?

To determine the formula for the scalar triple product, the cross product of two vectors is calculated first. After that, the dot product of the remaining vector with the resultant vector is calculated. If the triple product results to be zero, then it suggests that one of the three vectors taken is of zero magnitudes.

How do you prove coplanar points?

Points that are located on a plane are coplanar 4 points are coplanar if the volume created by the points is 0. If any three points determine a plane then additional points can be checked for coplanarity by measuring the distance of the points from the plane, if the distance is 0 then the point is coplanar.

How do you know if a vector is coplanar?

If the scalar triple product of any three vectors is zero then they are coplanar. If any three vectors are linearly dependent then they are coplanar. n vectors are coplanar if among them no more than two vectors are linearly independent vectors.

Can three vectors add up to zero if they are not coplanar?

No, three non-coplanar vectors cannot ad up to given zero resultant because for non-coplanar vectors the resultant of the two vectors will not lie in the plane of third vector , and so the resultant cannot cancel the third vector to given null vector .

How do you prove that four points are coplanar?

A necessary and sufficient condition for four points A(a ),B(b ),C(c ),D(d ) to be coplanar is that, there exist four scalars x,y,z,t not all zero such that xa +yb +zc +td =0 and x+y+z+t=0.

What is scalar triple product of vector?

The scalar triple product of three vectors a, b, and c is (a×b)⋅c. The scalar triple product is important because its absolute value |(a×b)⋅c| is the volume of the parallelepiped spanned by a, b, and c (i.e., the parallelepiped whose adjacent sides are the vectors a, b, and c).

What is scalar triple product?

The scalar triple product (also called the mixed product, box product, or triple scalar product) is defined as the dot product of one of the vectors with the cross product of the other two.

How do you find coplanar lines?

Given two lines L1 and L2, each passing through a point whose position vector are given as (X, Y, Z) and parallel to line whose direction ratios are given as (a, b, c), the task is to check whether line L1 and L2 are coplanar or not. Coplanar: If two lines are in a same plane then lines can be called as coplanar.

What is the scalar triple product of three non-zero vectors?

Now we use scalar triple product for the characterisation of coplanar vectors. The scalar triple product of three non-zero vectors is zero if, and only if, the three vectors are coplanar. Let , , be any three non-zero vectors.

How do you prove that the vectors are coplanar?

iii) If the triple product of vectors is zero, then it can be inferred that the vectors are coplanar in nature. The triple product indicates the volume of a parallelepiped. If it is zero, then such a case could only arise when any one of the three vectors is of zero magnitude.

What is the scalar triple product of a parallelpiped?

If any two of the vectors are parallel then the parallelpiped collapses to a shape with zero volume. Because the scalar triple product denotes the volume of a parallelepid spanned by the three vectors a →, b → and c →. If two vectors coincide the parallelepid degenerates to a parallelogram with zero volume.

What is the triple product of co-Planer vectors?

The triple product also known as box product. In this case a box with volume zero Simply put… the scalar triple product is a measure of orthogonality (or some volume). So co-planer vectors lie on the same plane meaning there is zero orthogonality. At least that’s a way to simplify things.