Answer

Verified

387.3k+ views

**Hint:**Here, we are required to find the linear equation of the plane through the given point which contains the line with the given vector equation. Thus, we will use the parametric form of equation of line and substitute the given values and with the help of plane and vector equation, we will solve it further to find the required linear equation of the given plane.

**Formula Used:**

1. The parametric line equation is given by: $p = {p_0} + t \cdot \overrightarrow v $

2. The plane equation is given by: $\left\langle {\overrightarrow w ,p - {p_1}} \right\rangle = 0$

**Complete step-by-step answer:**

As we know,

The parametric line equation is given by:

$p = {p_0} + t \cdot \overrightarrow v $

Where, $p = \left\{ {x,y,z} \right\}$, ${p_0} = \left\{ {0,6,1} \right\}$ and $\overrightarrow v = \left\{ {3, - 2, - 2} \right\}$

Also, the plane equation is given by

$\left\langle {\overrightarrow w ,p - {p_1}} \right\rangle = 0$

Here, according to the question, we have, ${p_1} = \left\{ {1,2,3} \right\}$and $\overrightarrow w $ is perpendicular to $\overrightarrow v $ and to the segment ${p_1} - {p_0}$ such that:

$\overrightarrow w = \overrightarrow v \times \left( {{p_1} - {p_0}} \right) = \left\{ {3, - 2, - 2} \right\} \times \left\{ {1, - 4, - 2} \right\}$

Then,

$\overrightarrow w = \left\{ { - 4,4, - 10} \right\}$

The plane equation is given as:

$\left( { - 4} \right)\left( {x - 1} \right) + 4\left( {y - 2} \right) + \left( { - 10} \right)\left( {z - 3} \right) = 0$

$ \Rightarrow - 4x + 4 + 4y - 8 - 10z + 30 = 0$

Solving the like terms further, we get,

$ \Rightarrow - 4x + 4y - 10z + 26 = 0$

**Therefore, the linear equation of the plane through the point $\left( {1,2,3} \right)$ which contains the line represented by the vector equation $r\left( t \right) = \left\langle {3t,6 - 2t,1 - 2t} \right\rangle $ is represented by $ - 4x + 4y - 10z + 26 = 0$.**

Hence, this is the required answer.

Hence, this is the required answer.

**Note:**

Parametric equation is a type of equation that employs an independent variable called a parameter (often denoted by $t$) and in which dependent variables are defined as continuous functions of the parameter and are not dependent on another existing variable. Also, a vector equation is any function that takes any one or more variables and returns a vector. The vector equation of a line is an equation that identifies the position vector of every point along the line.

Recently Updated Pages

What number is 20 of 400 class 8 maths CBSE

Which one of the following numbers is completely divisible class 8 maths CBSE

What number is 78 of 50 A 32 B 35 C 36 D 39 E 41 class 8 maths CBSE

How many integers are there between 10 and 2 and how class 8 maths CBSE

The 3 is what percent of 12 class 8 maths CBSE

Find the circumference of the circle having radius class 8 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

One cusec is equal to how many liters class 8 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

Change the following sentences into negative and interrogative class 10 english CBSE