Vectors
Vectors
Allen Star Batch
Grade 12
Question:
The volume of a right triangular prism $ABC A_1 B_1 C_1$ is equal to 3. If the position vectors of the vertices of the base $ABC$ are $A(1, 0, 1); B(2,0, 0)$ and $C(0, 1, 0)$ the position vectors of the vertex $A_1$ can be:
(2, 2, 2)
(0, 2, 0)
(0, -2, 2)
(0, -2, 0)
Step-by-Step Solution
Key Concept: For a right prism, the lateral edge AA₁ must be perpendicular to the base plane ABC. Given volume = 3 and base area = ½|AB × AC|, find altitude H, then A₁ = A + H·n̂ where n̂ is the unit normal to the base plane.
Given the prism volume, calculate altitude $H = (AA_1) = \sqrt{6}$. Let vertex $A_1(x, y, z)$ and use $\vec{AA_1} = (x-1, y, z-1)$. Apply the perpendicularity condition $\vec{AA_1} \perp \vec{AC}$ and compute the unit normal vector to the base using $\vec{n} = \frac{\vec{AB} \times \vec{AC}}{|\vec{AB} \times \vec{AC}|}$. From the given normal $\sqrt{6}\vec{n} = (\vec{i} + 2\vec{j} + \vec{k})$, extract the coordinates and verify using both perpendicularity and magnitude conditions.
Correct Answer: 1,4