Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 12

Find the Vector and Cartesian Equations of the Plane that Passes Through the Point (5, 2, −4) and is Perpendicular to the Line with Direction Ratios 2, 3, −1. - Mathematics

Sum

Find the vector and Cartesian equations of the plane that passes through the point (5, 2, −4) and is perpendicular to the line with direction ratios 2, 3, −1.

Solution

  \text{ We know that the vector equation of the plane passing through a point} \vec{a} \text{ and normal to }\vec{n}  \text{ is }

$\vec{r} . \vec{n} = \vec{a} . \vec{n}$

 \text{ Substituting }\vec{a} = \text{ 5 }\hat{i }+ \text{ 2 }\hat{j } - \text{ 4 }\hat{k } \text{ and }\vec{n} = \text{ 2 }\hat{i } + \text{ 3 }\hat{j } - \hat{k  }     (\text{ because the direction ratios of  } \vec{n} \text{  are 2, 3, -1)}, \text{ we get }

 \vec{r} . (\text{ 2 }\hat{i} + \text{ 3 }\hat{  j } - \hat{  k})= \text{ 5 }\hat{  i }+ \text{ 2 }\hat{  j } - \text{ 4 }\hat{  k } \text{ and }\vec{n}. ( 2 \hat{i }+3 \hat{j }- \hat{k } )

  ⇒  \vec{r} . (\text{ 2 }\hat{  i } + \text{ 3 }\hat{  j } - \hat{  k}) = 10 + 6 + 4

  ⇒   \vec{r} . (\text{ 2 }\hat{  i } + \text{ 3 }\hat{  j } - \hat{  k}) = 20

 \text { For Cartesian form, we need to substitute } \vec{r} = \vec{r} . (\text{ x }\hat{  i } + \text{ y }\hat{  j } - \text{ z }\hat{  k}) \text{ in this equation. Then, we get }

 (\text{ x }\hat{i } + \text{ y }\hat{j }  + \text{ z }\hat{k }) .(\text{ 2 }\hat{i } + \text{ 3 }\hat{ j } - \hat{k})= 20

$\Rightarrow 2x + 3y - z = 20$

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APPEARS IN

RD Sharma Class 12 Maths
Chapter 29 The Plane
Exercise 29.3 | Q 18 | Page 14