Home | | **Maths 12th Std** | Equation of a plane passing through two given distinct points and is parallel to a non-zero vector

(a) Parametric form of vector equation (b) Non-parametric form of vector equation (c) Cartesian form of equation

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

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

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

Where ( - ) and * *are not parallel vectors.

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, âˆ’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, âˆ’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 = *x*Ë†*i + y*Ë†*j + z*Ë†*k* 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 .

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

The required plane is parallel to the given line and so it is
parallel to the vector * *= *i*Ë† + Ë†*j *âˆ’ Ë†*k *and
he plane passes through the points * *= âˆ’Ë†*i *+ 2Ë†*j*, = 2Ë†*i *+ 2Ë†*j *âˆ’ Ë†*k *.

vector equation of the plane in parametric form is , where
s, t âˆˆ **R** which implies that = (-Ë†*i* + 2Ë†*j* ) + s (3Ë†*i* - Ë†*k* )+ t (Ë†*i* + Ë†*j* - Ë†*k* ) , where s, t âˆˆ **R**.

vector equation of the plane in non-parametric form is

we have ( - (*-i*Ë†* + 2*Ë†* j)) . (*Ë†*i + 2*Ë†*j + 3*Ë†*k ) = 0 **â‡’** ** . (*Ë†*i + 2*Ë†* j + 3*Ë†*k )* = 3

If = *x*Ë†*i + y*Ë†*j + z*Ë†*k* 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 .

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12th Mathematics : UNIT 6 : Applications of Vector Algebra : Equation of a plane passing through two given distinct points and is parallel to a non-zero vector |

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