Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

Let ${\vec{n}}_1$ be the normal vector to the plane $P_1$ containing vectors $\vec{a}$ and $\vec{b}$.

${\vec{n}}_1=\vec{a}\times \vec{b}$

Let ${\vec{n}}_2$ be the normal vector to the plane $P_2$ containing vectors $\vec{c}$ and $\vec{d}$.

${\vec{n}}_2=\vec{c}\times \vec{d}$

It is given that $(\vec{a}\times \vec{b})\times (\vec{c}\times \vec{d})=\vec{0}$.

Substituting the normal vectors, we get:

${\vec{n}}_1\times {\vec{n}}_2=\vec{0}$

This implies that the normal vectors ${\vec{n}}_1$ and ${\vec{n}}_2$ are parallel.

Since the normals to the planes are parallel, the planes $P_1$ and $P_2$ are also parallel.

Thus, the angle between the planes $P_1$ and $P_2$ is $0$.

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