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:
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:
Show Hint
- Ratio of areas of the triangles formed by \(\mathbf{0}, \mathbf{a}, \mathbf{b}\) to \(\mathbf{0}, \mathbf{b}, \mathbf{c}\)
- Ratio of the numerical values of volume of the parallelepiped formed with \(\mathbf{0}, \mathbf{a}, \mathbf{b}, \mathbf{c}\) and its height
- Ratio of lengths of the diagonals of the parallelepiped formed with \(\mathbf{0}, \mathbf{a}, \mathbf{b}, \mathbf{c}\)
- Length of the perpendicular from origin to the plane
The Correct Option is D
Solution and Explanation
\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 \(\mathbf{a}, \mathbf{b}, \mathbf{c}\).
Top Questions on Geometry and Vectors
If \( \vec{u}, \vec{v}, \vec{w} \) are non-coplanar vectors and \( p, q \) are real numbers, then the equality:
\[ [3\vec{u} \quad p\vec{v} \quad p\vec{w}] - [p\vec{v} \quad \vec{w} \quad q\vec{u}] - [2\vec{w} \quad q\vec{v} \quad q\vec{u}] = 0 \]
holds for:
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and
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Given vectors \(\mathbf{a} = \mathbf{i} + \mathbf{j} - 2\mathbf{k}\), \(\mathbf{b} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k}\), \(\mathbf{c} = 2\mathbf{i} - \mathbf{j} + \mathbf{k}\), and \(\mathbf{r}\) such that
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- Let \( \vec{a} = 2\hat{i} + \hat{j} + 3\hat{k} \), \( \vec{b} = 3\hat{i} + 3\hat{j} + \hat{k} \), and \( \vec{c} = \hat{i} - 2\hat{j} + 3\hat{k} \) be three vectors. If \( \vec{r} \) is a vector such that \( \vec{r} \times \vec{a} = \vec{r} \times \vec{b} \) and \( \vec{r} . \vec{c} = 18 \), then the magnitude of the orthogonal projection of \( 4\hat{i} + 3\hat{j} - \hat{k} \) on \( \vec{r} \) is:
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