Question

2. Let vectors A = (2, 1, 3), B = (-5, 2, 1), and C = (2, 1, 1). Find the volume of the parallelepiped defined by vectors A, B, and C by calculating A ● (B x C) using the dot and cross product rules.

Answer 1

Answer:

Step-by-step explanation:

Volume of the parallelopiped

=[a b c]

$=\left|\begin{array}{ccc}2&1&3\\-5&2&1\\2&1&1\end{array}\right|$

=2(2-1)-1(-5-2)+3(-5-4)

= 2+7-27 = -18

But the volume cannot be negative.

Therefore volume of the parallelopiped is

18 cubic units

Answer 2

Answer:

So, volume of our parallelopiped is -18

Step-by-step explanation:

Our given vectores are

OA = 2i + j + 3k

OB = -5i + 2j + k

OC = 2i + j + k

As we know,

Volume of parallelopiped = OA . (OB x OC)

Lets calculate OB x OC first using the determinents

$OB\times OC = \left[\begin{array}{ccc}i&j&k\\-5&2&1\\2&1&1\end{array}\right] \\Expanding\\\\OB\times OC=i (2\times1 - 1\times1) - j (-5\times1-2\times1) + k(-5\times1-2\times2)\\\\OB\times OC=i (2 - 1) - j (-5-2) + k(-5-4)\\\\OB\times OC=i (1) - j (-7) + k(-9)\\\\OB\times OC=i +7j -9k$

Now,lets take dot product of $OB\times OC$ with OA

$OA.(OB\times OC)=(2i + j + 3k).(i +7j -9k)$

$OA.(OB\times OC)=2(i.i) + 7(j.j) + 3\times-9(k.k)$

$OA.(OB\times OC)=2 + 7 - 27$

$OA.(OB\times OC)=-18$

So, volume of our parallelopiped is -18