Vector Algebra
Vectors
star_batch_jee_advanced_2025
Grade 12
Question:
Let $A, B, C$ be points with position vectors $\vec{r_1} = 2\hat{i} - \hat{j} + \hat{k}, \vec{r_2} = \hat{i} + 2\hat{j} + 3\hat{k}$ and $\vec{r_3} = 3\hat{i} + \hat{j} + 2\hat{k}$ relative to the origin 'O'. Find the shortest distance between point $B$ and plane $OAC$.
Step-by-Step Solution
Key Concept: The perpendicular distance from a point to a plane is found using the formula $d = \frac{|\vec{a} \cdot \vec{n} - d|}{|\vec{n}|}$ where $\vec{n}$ is the normal vector.
The equation of plane OAC is found using the vector equation $(\vec{r} - \vec{0}) = \vec{r_1} \times (-\text{say } \pi)$. To find the perpendicular distance of point B from plane $\pi$, we use the standard distance formula with the plane equation.
Correct Answer: I need to find the shortest distance from point B to the plane OAC.
**Step 1: Find the equation of plane OAC**
The plane passes through origin O, so its equation is of the form $\vec{r} \cdot \vec{n} = 0$, where $\vec{n}$ is the normal vector.