Let the position vectors of the vertices A, B, and C of a tetrahedron ABCD be i + 2j + k, i + 3j - 2k, and 2i…

Question

Let the position vectors of the vertices A, B, and C of a tetrahedron ABCD be $\hat{i} + 2\hat{j} + \hat{k}$ , $\hat{i} + 3\hat{j} - 2\hat{k}$ , and $2\hat{i} + \hat{j} - \hat{k}$ respectively. The altitude from the vertex D to the opposite face ABC meets the median line segment through A of the triangle ABC at the point E. If the length of AD is $\frac{\sqrt{110}}{3}$ and the volume of the tetrahedron is $\frac{\sqrt{805}}{6\sqrt{2}}$ , then the position vector of E is:

A
$\frac{1}{2}(\hat{i} + 4\hat{j} + 7\hat{k})$
B
$\frac{1}{12}(7\hat{i} + 4\hat{j} + 3\hat{k})$
C
$\frac{1}{6}(12\hat{i} + 12\hat{j} + \hat{k})$
D
$\frac{1}{6}(7\hat{i} + 12\hat{j} + \hat{k})$

Accepted Answer

D. $$\frac{1}{6}(7\hat{i} + 12\hat{j} + \hat{k})$$

Answer

A. $$\frac{1}{2}(\hat{i} + 4\hat{j} + 7\hat{k})$$

Answer

B. $$\frac{1}{12}(7\hat{i} + 4\hat{j} + 3\hat{k})$$

Answer

C. $$\frac{1}{6}(12\hat{i} + 12\hat{j} + \hat{k})$$

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