Question

If $[\vec{a} \vec{b} \vec{c}]=4$, then the volume (in cubic units) of the parallelopiped with $\vec{a}+2\vec{b}, \vec{b}+2\vec{c}$ and $\vec{c}+2\vec{a}$ as coterminal edges, is

• A. 32
• B. 16
• C. 9
• D. 36

Hint:

A standard shortcut for the scalar triple product of cyclically shifted combinations like $[\vec{a}+k\vec{b}, \vec{b}+k\vec{c}, \vec{c}+k\vec{a}]$ is that it always equals $(1 + k^3)[\vec{a} \vec{b} \vec{c}]$. Here $k=2$, so $(1 + 2^3) = 9$. Multiplying by 4 gives 36 in seconds!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given the scalar triple product (volume of a parallelopiped) of three vectors $\vec{a}, \vec{b}, \vec{c}$. We need to find the volume of a new parallelopiped formed by linear combinations of these vectors.

Step 2: Key Formula or Approach:
The volume of a parallelopiped with coterminal edges $\vec{x}, \vec{y}, \vec{z}$ is given by their scalar triple product $[\vec{x} \vec{y} \vec{z}]$.
For vectors formed by linear combinations, we can use determinant properties to factor out the coefficients relative to the original vectors:
$$[x_1\vec{a}+y_1\vec{b}+z_1\vec{c}, x_2\vec{a}+y_2\vec{b}+z_2\vec{c}, x_3\vec{a}+y_3\vec{b}+z_3\vec{c}] = \begin{vmatrix} x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ x_3 & y_3 & z_3 \end{vmatrix} [\vec{a} \vec{b} \vec{c}]$$

Step 3: Detailed Explanation:
The new edges are:
$\vec{v}_1 = 1\vec{a} + 2\vec{b} + 0\vec{c}$
$\vec{v}_2 = 0\vec{a} + 1\vec{b} + 2\vec{c}$
$\vec{v}_3 = 2\vec{a} + 0\vec{b} + 1\vec{c}$
Set up the determinant of their coefficients:
$$V' = \begin{vmatrix} 1 & 2 & 0 \\ 0 & 1 & 2 \\ 2 & 0 & 1 \end{vmatrix} [\vec{a} \vec{b} \vec{c}]$$
Expand the determinant along the first row:
$$\Delta = 1(1 \times 1 - 2 \times 0) - 2(0 \times 1 - 2 \times 2) + 0$$
$$\Delta = 1(1) - 2(-4)$$
$$\Delta = 1 + 8 = 9$$
So, the new volume $V'$ is $9 \times [\vec{a} \vec{b} \vec{c}]$.
We are given that $[\vec{a} \vec{b} \vec{c}] = 4$.
$$V' = 9 \times 4 = 36$$

Step 4: Final Answer:
The volume of the new parallelopiped is 36 cubic units, matching option (D).