First slide

Planes in 3D

Question

$The vector equation of the plane passing through the origin and the line of intersection of the planes \vec{r} . \vec{a} = λ and \vec{r} \cdot \vec{b} = μ is$

Moderate

A

$\vec{r} \cdot (λ \vec{a} - μ \vec{b}) = 0$
B

$\vec{r} \cdot (λ \vec{b} - μ \vec{a}) = 0$
C

$\vec{r} \cdot (λ \vec{a} + μ \vec{b}) = 0$
D

$\vec{r} \cdot (λ \vec{b} + μ \vec{a}) = 0$

Solution

The equation of a plane through the line of intersection of the
$planes \vec{r} \cdot \vec{a} = λ and \vec{r} \cdot \vec{b} = μ is$
$\begin{array}{l} (\vec{r} \cdot \vec{a} - λ) + k (\vec{r} \cdot \vec{b} - μ) = 0 \\ or \vec{r} \cdot (\vec{a} + k \vec{b}) = λ + kμ - - - - - - - - (i) \end{array}$
Putting the value of k in (i), we get the equation of the required plane as
$\vec{r} \cdot (μ \vec{a} - λ \vec{b}) = 0 or \vec{r} \cdot (λ \vec{b} - μ \vec{a}) = 0$

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