Coplanarity of Two Lines

Lines that are coplanar are those that lay on the same plane as one another. 

Use the condition both in vector form and in Cartesian form to demonstrate that two lines lie on the same plane.

In three-dimensional geometry, a typical issue that is covered is that of coplanar lines. 

Coplanarity is a situation that arises in mathematical theory when a certain number of lines all lie on the same plane. 

When this occurs, we say that the lines are coplanar with one another.

To refresh your memory, a plane is a two-dimensional figure that extends into infinity in a three-dimensional space, 

whereas we have utilised vector equations to depict straight lines in this discussion (also referred to as lines).

Proof

Let us consider the points (-1, 3) and (-1), respectively (2,-3).

The equation of a line that passes through the points (x₁, y₁) and (x₂, y₂) is given by the expression y – y₁ = m, as is common knowledge (x – x₁).

The slope, denoted by the symbol m, is calculated as follows: m = (y₂ – y₁) / (x₂ – x₁)

Utilizing Cuemath’s Slope Calculator will allow you to determine the slope.

Consequently, by plugging the supplied points into the equation of a line, we obtain:

y – 1 = m (x – (-1)) ———-(1)

m = (y₂ – y₁) / (x₂ – x₁)

m = (-3 – 3) / (2 – (-1))

m = -6 / 3 = -2

When we plug in the value that we have for m into equation (1), we get

y – 3 = -2 (x + 1)

y – 3 = -2x – 2 

y = -2x + 1 

2x + y = 1

Since this is the case, the equation of the line that contains the points (-1, 3) and (2, -3) is 2x + y = + 1.

Position vector

The position vector is a straight line that is used to define the location of a moving point in relation to a body.

One end of the line is attached to the body, while the other end of the line is attached to the moving point. 

When the point moves, the position vector will undergo a change in either its length or its direction, or both, depending on the nature of the movement.

Formula for Positioning Using a Vector

We can use a formula to determine a position vector between any two points in the xy plane if we know the positions of the points themselves and any point in the xy plane. 

Take, for example, two points A and B, one of which has the coordinates (xk, yk) in the xy-plane, and the other of which has the coordinates (xk+1, yk+1). 

Both of these points lie in the same plane.

AB = (xk+1 – xk, yk+1 – yk) is the formula that can be used to obtain the position vector that extends from A to B.

The position vector AB is a vector that extends from point A all the way to point B. 

It begins at point A and finishes at point B.

If we wish to get the position vector from point B to point A, we may apply the following formula:

BA = (xk – xk+1, yk – yk+1)

Conclusion

We refer to the lines that are parallel to one another on the same plane as the coplanar lines.

In the context of three-dimensional geometry, the idea of coplanar lines is a fundamental one. 

Remember that a plane is a two-dimensional figure that extends into infinity in three-dimensional space, and that in order to depict lines or straight lines, vector equations have been put into practise here. 

A plane, on the other hand, is a figure that only has two dimensions. 

In light of this, the objective is to determine whether or not two lines, L1 and L2, which each pass through a point whose position 

vector is given as (A, B, C) and which are parallel to a line whose direction ratios are given as (X, Y, Z), are coplanar. 

This can be done by examining the relationship between the two lines.