The length of the perpendicular from the origin to the plane passing through the point $\vec{a}$ and containing the line $\vec{r} = \vec{b} + λ \vec{c}$ is

Moderate

A

$\frac{[\vec{a} \vec{b} \vec{c}]}{| \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} |}$
B

$\frac{[\vec{a} \vec{b} \vec{c}]}{| \vec{a} \times \vec{b} + \vec{b} \times \vec{c} |}$
C

$\frac{[\vec{a} \vec{b} \vec{c}]}{\vec{∣ b} \times \vec{c} + \vec{c} \times \vec{a} ∣}$
D

$\frac{[\vec{a} \vec{b} \vec{c}]}{| \vec{c} \times \vec{a} + \vec{a} \times \vec{b} |}$

Solution

The given plane passes through $\vec{a}$ and is parallel to the vectors $\vec{b} - \vec{a}$ and $\vec{c}$ So it is normal to
$(\vec{b} - \vec{a}) \times \vec{c}$ Hence, its equation is
$(\vec{r} - \vec{a}) \cdot ((\vec{b} - \vec{a}) \times \vec{c}) = 0$
or $\vec{r} \cdot (\vec{b} \times \vec{c} + \vec{c} \times \vec{a}) = [\vec{a} \vec{b} \vec{c}]$
The length of the perpendicular from the origin to this plane is
$\frac{[\vec{a} \vec{b} \vec{c}]}{| \vec{b} \times \vec{c} + \vec{c} \times \vec{a} |}$

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