Volume of tetrahedron whose vertices are the points with position vectors $\hat{i} - 6\hat{j} + 10\hat{k}, -\hat{i} - 3\hat{j} + 7\hat{k}, 5\hat{i} - \hat{j} + h\hat{k}$ and $7\hat{i} - 4\hat{j} + 7\hat{k}$ is $11$ cubic units then the value of $h$ is ______ ($h > 1$)
Step-by-Step Solution
Key Concept: The volume of a tetrahedron is $\frac{1}{6}$ times the absolute value of the scalar triple product of three edge vectors from one vertex.
Given position vectors $\vec{r_1} = \vec{i} - 6\vec{j} + 10\vec{k}$, $\vec{r_2} = -\vec{i} - 3\vec{j} + 7\vec{k}$, $\vec{r_3} = 5\vec{i} - \vec{j} + h\vec{k}$, $\vec{r_4} = -\vec{i} - 4\vec{j} + 7\vec{k}$, the edge vectors are computed as $\vec{a} = \vec{r_1} - \vec{r_2} = 2\vec{i} - 3\vec{j} + 3\vec{k}$, $\vec{b} = \vec{r_3} - \vec{r_2} = 6\vec{i} + 2\vec{j} + (h-7)\vec{k}$, $\vec{c} = \vec{r_4} - \vec{r_1} = 6\vec{i} + 2\vec{j} - 3\vec{k}$. The volume of tetrahedron is $\frac{1}{2}|\vec{a} \ \vec{b} \ \vec{c}| = 11$.
Correct Answer: I need to find the value of h using the tetrahedron volume formula.
**Step 1: Set up the scalar triple product**
The volume of a tetrahedron is given by:
$$V = \frac{1}{6}|[\vec{a} \cdot (\vec{b} \times \vec{c})]|$$