NON-COPLANAR VECTORS: DEFINITION, CONDITIONS, EXERCISES - PHYSICAL - 2025

The non - coplanar vectors are those that do not share the same plane. Two free vectors and a point define a single plane. A third vector may or may not share that plane, and if it does not, they are non-coplanar vectors.

Non-coplanar vectors cannot be represented in two-dimensional spaces like a blackboard or sheet of paper, because some of them are contained in the third dimension. To represent them properly you have to use perspective.

Figure 1. Coplanar and non-coplanar vectors. (Own elaboration)

If we look at figure 1, all the objects shown are strictly in the plane of the screen, however thanks to perspective our brain is able to imagine a plane (P) that comes out of it.

On that plane (P) are the vectors r, s, u, while the vectors v and w are not in that plane.

Therefore the vectors r, s, u are coplanar or coplanar to each other since they share the same plane (P). Vectors v and w do not share a plane with any of the other vectors shown, therefore they are non-coplanar.

Coplanar Vectors and Equation of the Plane

A plane is uniquely defined if there are three points in three-dimensional space.

Suppose those three points are point A, point B, and point C that define the plane (P). With these points it is possible to construct two vectors AB = u and AC = v that are by construction coplanar with the plane (P).

The vector product (or cross product) of these two vectors results in a third vector perpendicular (or normal) to both of them and therefore perpendicular to the plane (P):

n = u X v => n ⊥ u and n ⊥ v => n ⊥ (P)

Any other point that belongs to the plane (P) must satisfy that the vector AQ is perpendicular to the vector n; This is equivalent to saying that the dot product (or dot product) of n with AQ must be zero:

n • AQ = 0 (*)

The previous condition is equivalent to saying that:

AQ • (u X v) = 0

This equation ensures that point Q belongs to plane (P).

Cartesian equation of the plane

The above equation can be written in Cartesian form. To do this, we write the coordinates of the points A, Q and the components of the normal vector n:

So the components of AQ are:

The condition for the vector AQ to be contained in the plane (P) is the condition (*) which is now written like this:

Calculating the dot product remains:

If it is developed and rearranged it remains:

The previous expression is the Cartesian equation of a plane (P), as a function of the components of a vector normal to (P) and the coordinates of a point A that belongs to (P).

Conditions for three vectors to be non-coplanar

As seen in the previous section, the condition AQ • (u X v) = 0 guarantees that the vector AQ is coplanar to u and v.

If we call the vector AQ w then we can affirm that:

w, u and v are coplanar, if and only if w • (u X v) = 0.

Non-coplanarity condition

If the triple product (or mixed product) of three vectors is different from zero then those three vectors are non-coplanar.

If w • (u X v) ≠ 0 then the vectors u, v, and w are non-coplanar.

If the Cartesian components of the vectors u, v, and w are introduced, the condition of non-coplanarity can be written like this:

The triple product has a geometric interpretation and represents the volume of the parallelepiped generated by the three non-coplanar vectors.

Figure 2. Three non-coplanar vectors define a parallelepiped whose volume is the module of the triple product. (Own elaboration)

The reason is as follows; When two of the non-coplanar vectors are multiplied vectorially, a vector is obtained whose magnitude is the area of ​​the parallelogram that they generate.

Then when this vector is scalarly multiplied by the third non-coplanar vector, what we have is the projection to a vector perpendicular to the plane that the first two determine multiplied by the area that they determine.

In other words, we have the area of ​​the parallelogram generated by the first two multiplied by the height of the third vector.

Alternative condition of non-coplanarity

If you have three vectors and any of them cannot be written as a linear combination of the other two, then the three vectors are non-coplanar. That is, three vectors u, v and w are non-coplanar if the condition:

α u + β v + γ w = 0

It is only satisfied when α = 0, β = 0 and γ = 0.

Solved exercises

-Exercise 1

There are three vectors

u = (-3, -6, 2); v = (4, 1, 0) and w = (-1, 2, z)

Note that the z component of the vector w is unknown.

Find the range of values ​​that z can take such that the three vectors are guaranteed not to share the same plane.

Solution

w • (u X v) = -3 (z - 0) + 6 (4 z - 0) + 2 (8 + 1) = -3z + 24z + 18 = 21z + 18

We set this expression equal to the value zero

21 z + 18 = 0

and we solve for z

z = -18 / 21 = -6/7

If the variable z took the value -6/7 then the three vectors would be coplanar.

So the values ​​of z that guarantee that the vectors are non-coplanar are those in the following interval:

z ∈ (-∞, -6 / 7) U (-6/7, ∞)

-Exercise 2

Find the volume of the parallelepiped shown in the following figure:

Solution

To find the volume of the parallelepiped shown in the figure, the Cartesian components of three concurrent non-coplanar vectors at the origin of the coordinate system will be determined. The first one is the vector u of 4m and parallel to the X axis:

u = (4, 0, 0) m

The second is the vector v in the XY plane of size 3m that forms 60º with the X axis:

v = (3 * cos 60º, 3 * sin 60º, 0) = (1.5, 2.6, 0.0) m

And the third is the vector w of 5m and whose projection in the XY plane forms 60º with the X axis, in addition w forms 30º with the Z axis.

w = (5 * sin 30º * cos 60º, 5 * sin 30º * sin 60º, 5 * sin 30º)

Once the calculations have been carried out, we have: w = (1.25, 2.17, 2.5) m.

References

1. Figueroa, D. Series: Physics for Sciences and Engineering. Volume 1. Kinematics. 31-68.
2. Physical. Module 8: Vectors. Recovered from: frtl.utn.edu.ar
3. Hibbeler, R. 2006. Mechanics for Engineers. Static 6th Edition. Continental Publishing Company. 28-66.
4. McLean, W. Schaum Series. Mechanics for Engineers: Statics and Dynamics. 3rd Edition. McGraw Hill. 1-15.
5. Wikipedia. Vector. Recovered from: es.wikipedia.org