Question

Let \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) be position vectors of three non-collinear points on a plane. If

\[ \alpha = \left[\mathbf{a} \quad \mathbf{b} \quad \mathbf{c}\right] \text{ and } \mathbf{r} = \mathbf{a} \times \mathbf{b} - \mathbf{c} \times \mathbf{b} - \mathbf{a} \times \mathbf{c}, \]

Then \(\frac{|\alpha|}{|\mathbf{r}|}\) represents:

• A. Ratio of areas of the triangles formed by \(\mathbf{0}, \mathbf{a}, \mathbf{b}\) to \(\mathbf{0}, \mathbf{b}, \mathbf{c}\)
• B. Ratio of the numerical values of volume of the parallelepiped formed with \(\mathbf{0}, \mathbf{a}, \mathbf{b}, \mathbf{c}\) and its height
• C. Ratio of lengths of the diagonals of the parallelepiped formed with \(\mathbf{0}, \mathbf{a}, \mathbf{b}, \mathbf{c}\)
• D. Length of the perpendicular from origin to the plane

Hint:

Scalar triple product gives volume, and the magnitude of vector cross products relate to area and normals, useful in distance calculations.

Answer 1

The Correct Option is D

\(\alpha = [\mathbf{a}
\mathbf{b}
\mathbf{c}]\) represents the scalar triple product (volume of parallelepiped formed by \(\mathbf{a}, \mathbf{b}, \mathbf{c}\)). \[ \mathbf{r} = \mathbf{a} \times \mathbf{b} - \mathbf{c} \times \mathbf{b} - \mathbf{a} \times \mathbf{c}. \] Simplifying \(\mathbf{r}\) shows it is proportional to the normal vector to the plane formed by the vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c}\). Hence, the ratio \(\frac{|\alpha|}{|\mathbf{r}|}\) gives the perpendicular distance from the origin to the plane defined by \(\mathbf{a}, \mathbf{b}, \mathbf{c}\).