List of top Mathematics Questions on 3D Geometry asked in UP Board XII

Find the vector equation of a plane which passes through the point of intersection of the planes \[ \vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 6 \, \text{and} \, \vec{r} \cdot (2\hat{i} + 3\hat{j} + 4\hat{k}) = -5 \] and the point $(1,1,1)$.

Find the equation of the plane which passes through the intersecting point of the planes \[ \vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 6 \, \text{and} \, \vec{r} \cdot (2\hat{i} + 3\hat{j} + 4\hat{k}) = -5, \] and the point \( (1, 1, 1) \).

Find the shortest distance between two lines: \[ \mathbf{r_1} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda (2\hat{i} + 3\hat{j} + 6\hat{k}) \] and \[ \mathbf{r_2} = 3\hat{i} + 3\hat{j} - 5\hat{k} + \mu (2\hat{i} + 3\hat{j} + 6\hat{k}) \]
Find the shortest distance between the lines \[ \vec{r_1} = (1 - t) \hat{i} + (t - 2) \hat{j} + (3 - 2t) \hat{k} \] \text{and} \[ \vec{r_2} = (s + 1) \hat{i} + (2s - 1) \hat{j} - (2s + 1) \hat{k}. \]
Find the shortest distance between the lines \[ \vec{r} = \hat{i} + 2\hat{j} + 3\hat{k} + \lambda(\hat{i} - 3\hat{j} + 2\hat{k}) \] and \[ \vec{r} = 4\hat{i} + 5\hat{j} + 6\hat{k} + \mu(2\hat{i} + 3\hat{j} + \hat{k}). \]
Find the vector and Cartesian equation of the plane which passes through the point $(1,0,-2)$ and on which the vector $\hat{i} + \hat{j} - \hat{k}$ is perpendicular.
Find the distance of the point $(2,5,-3)$ from the plane \[ \vec{r}\cdot(6\hat{i}-3\hat{j}+2\hat{k})=4. \]
Find the shortest distance between the lines \[ \vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k}) \] \[ \vec{r} = 3\hat{i} + 3\hat{j} - 5\hat{k} + \mu(2\hat{i} + 3\hat{j} + 6\hat{k}) \]
Find the vector equation of the line $\dfrac{x+3}{2} = \dfrac{y-5}{4} = \dfrac{z+6}{2}$.