First slide

Application of vectors in geometry

Question

$Equation of the plane passing through the line of intersection of the two planes \vec{r} \cdot {\vec{n}}_{1} = q_{1} and$
$\vec{r} \cdot {\vec{n}}_{2} = q_{2} and parallel to the line of intersection of \vec{r} \cdot {\vec{n}}_{3} = q_{3} and \vec{r} \cdot {\vec{n}}_{4} = q_{4} is$

Difficult

A

$dependent on {\vec{n}}_{1} \cdot {\vec{n}}_{3}$
B

$dependent on {\vec{n}}_{3} \cdot {\vec{n}}_{4}$
C

$independent of q_{1} and q_{2}$
D

$independent of q_{3} and q_{4}$

Solution

Equation of the required plane is

$\begin{array}{l} (\bar{r} \cdot {\bar{n}}_{1} - q_{1}) + λ (\bar{r} \cdot {\bar{n}}_{2} - q_{2}) = 0 \\ \Rightarrow \bar{r} \cdot ({\bar{n}}_{1} + λ n_{2}^{-}) = (q_{1} + λ q_{2}) \\ Now, perpendicular to {\bar{n}}_{3} \times {\bar{n}}_{4} \\ \Rightarrow ({\bar{n}}_{1} + λ {\bar{n}}_{2}) \cdot ({\bar{n}}_{3} \times {\bar{n}}_{4}) = 0 \end{array}$
$So independent on q_{3} and q_{4} only.$

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