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Three Dimensional Geometry — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSThree Dimensional Geometry2 Marks
Question
Find the cartesian form of the equation of a plane whose vector equation is:
$\vec{\text{r}}\cdot\big(12\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}}\big)+5=0$ 
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Answer

Given the vector equation of a plane,
$\vec{\text{r}}\cdot\big(12\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}}\big)+5=0$
Let, $\vec{\text{r}}=\big(\text{x}\hat{\text{i}}+\text{y}\hat{\text{j}}+\text{z}\hat{\text{k}}\big)$
$\big(\text{x}\hat{\text{i}}+\text{y}\hat{\text{j}}+\text{z}\hat{\text{k}}\big)\big(12\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}}\big)+5=0$
$(\text{x})(12)+(\text{y})(-3)+(\text{z})(4)=0$
$12\text{x}-3\text{y}+4\text{z}+5=0$
Cartesian form of the equation of the plane of the plane is given by
$12\text{x}-3\text{y}+4\text{z}+5=0$

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