$A = (0, 1, 2), B = (3, 0, 1), C = (4, 3, 6), D = (2, 3, 2)$ are the rectangular cartesian co-ordinates. Find the volume of the tetrahedron ABCD.

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Solution

The correct option is B 6
Let
$A = (0, 1, 2)$
$B = (3, 0, 1)$
$C = (4, 3, 6)$
$D = (2, 3, 2)$

Then
$A B = ¯ a = (3 - 0) i + (0 - 1) j + (1 - 2) k = 3 i - j - k$
$B C = ¯ b = (4 - 3) i + (3 - 0) j + (6 - 1) k = i + 3 j + 5 k$
$C D = ¯ c = (2 - 4) i + (3 - 3) j + (2 - 6) k = - 2 i + 0 j - 4 k$

Now
$(¯ a \times ¯ b) = (3 i - j - k) \times (i + 3 j + 5 k) = - 2 i - 16 j + 10 k$

Now
$¯ c . (¯ a \times ¯ b)$
$= (- 2 i + 0 j - 4 k) . (- 2 i - 16 j + 10 k)$
$= 4 + 56 - 40$
$= - 36$

Thus the required volume is

$= \frac{| - 36 |}{6}$

$= 6$ cubic unit.

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A = (0, 1, 2), B = (3, 0, 1), C = (4, 3, 6), D = (2, 3, 2)

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