Question

If $[a \ b \ c] = 4$ then volume of parallelopiped with coterminous edges $a+2b, \ b+2c, \ c+2a$ is

• A. 40 units
• B. 36 units
• C. 32 units
• D. 20 units

Answer 1

Answer: (B)

36 units

Answer 2

To find the volume of the parallelepiped, we need to compute the scalar triple product of the coterminous edges $a+2b$, $b+2c$, and $c+2a$.

The volume $V$ is given by:

$V = |[ (a+2b) \ (b+2c) \ (c+2a) ]|$

Expanding the scalar triple product:

$V = | (a+2b) \cdot ((b+2c) \times (c+2a)) |$

$V = | (a+2b) \cdot (b \times c + 2b \times a + 2c \times c + 4c \times a) |$

Since $c \times c = 0$, we have:

$V = | (a+2b) \cdot (b \times c + 2b \times a + 4c \times a) |$

$V = | a \cdot (b \times c) + 2a \cdot (b \times a) + 4a \cdot (c \times a) + 2b \cdot (b \times c) + 4b \cdot (b \times a) + 8b \cdot (c \times a) |$

Using the properties of scalar triple products, terms like $a \cdot (b \times a)$, $a \cdot (c \times a)$, $b \cdot (b \times c)$, and $b \cdot (b \times a)$ are all zero. Thus,

$V = | a \cdot (b \times c) + 8b \cdot (c \times a) |$

Since $b \cdot (c \times a) = a \cdot (b \times c)$, we have:

$V = | [a \ b \ c] + 8 [a \ b \ c] |$

$V = | 9 [a \ b \ c] |$

Given that $[a \ b \ c] = 4$,

$V = | 9 \times 4 | = 36$

Thus, the volume of the parallelepiped is 36 cubic units.