Let $A(2\hat{i}+3\hat{j}+5\hat{k}), B(-\hat{i}+3\hat{j}+2\hat{k})$ and $C(\hat{i}+5\hat{j}+\mu\hat{k})$ are vertices of a triangle and its median through $A$ is equally inclined to the positive directions of the axes. Find the value of $2\lambda - \mu$
Step-by-Step Solution
Key Concept: Equal inclination to all three axes means the direction cosines are equal: $\frac{1}{\sqrt{3}}$ each.
The midpoint of BC is $\left(\frac{x-1}{2}\right)\vec{i} + j + \left(\frac{x+2}{2}\right)\vec{k}$. Since this point is equally inclined to all coordinate axes, its direction cosines are $\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)$, meaning all components are equal.
Correct Answer: I need to find the value of 2λ - μ, but I notice the problem statement mentions λ is not defined in the given coordinates. Let me assume point A has coefficient λ instead of 2, and work through this systematically.
Given:
- A(λî + 3ĵ + 5k̂)
- B(-