Equation of a plane passing through two given distinct points and is parallel to a non-zero vector

Equation of a plane passing through two given distinct points and is parallel to a non-zero vector

(a) Parametric form of vector equation

The parametric form of vector equation of the plane passing through two given distinct points A and B with position vectors  and  , and parallel to a non-zero vector  is

(b) Non-parametric form of vector equation

Equation (1) can be written equivalently in non-parametric vector form as

Where (  - )  and  are not parallel vectors.

(c) Cartesian form of equation

This is the required Cartesian equation of the plane.

Example 6.43

Find the non-parametric form of vector equation, and Cartesian equation of the plane passing through the point (0,1, −5) and parallel to the straight lines 

Solution

We observe that the required plane is parallel to the vectors  and passing through the point (0,1, −5) with position vector . 

We observe that  is not parallel to . Then the vector equation of the plane in non-parametric form is given by  = 0.         â€¦(1)

If  = xˆi + yˆj + zˆk is the position vector of an arbitrary point on the plane, then from the above equation, we get the Cartesian equation of the plane as -9x + 8 y - z = 13 or 9x - 8 y + z +13 = 0 .

Example 6.44

Find the vector parametric, vector non-parametric and Cartesian form of the equation of the plane passing through the points (−1, 2, 0), (2, 2 −1) and parallel to the straight line

Solution

The required plane is parallel to the given line and so it is parallel to the vector  = iˆ + ˆj − ˆk and he plane passes through the points  = −ˆi + 2ˆj,  = 2ˆi + 2ˆj − ˆk .

vector equation of the plane in parametric form is , where s, t ∈ R which implies that  = (-ˆi + 2ˆj ) + s (3ˆi - ˆk )+ t (ˆi + ˆj - ˆk ) , where s, t ∈ R.

vector equation of the plane in non-parametric form is 

we have ( - (-iˆ + 2ˆ j)) . (ˆi + 2ˆj + 3ˆk ) = 0 ⇒  . (ˆi + 2ˆ j + 3ˆk ) = 3

If  = xˆi + yˆj + zˆk is the position vector of an arbitrary point on the plane, then from the above equation, we get the Cartesian equation of the plane as x + 2 y + 3z = 3 .