Question

Determine the volume of the parallelopiped determined by the vectors A=<−6,3,2> and B=<−2,10,−5> and C=<1,10,−3>.

Answer 1

Answer:

The volume of the parallelopiped is 213 cubic units.

Step-by-step explanation:

The given vectors are A=<−6,3,2> and B=<−2,10,−5> and C=<1,10,−3>.

If three vectors $a=<a_1,a_2,a_3>,b=<b_1,b_2,b_3>,c=<c_1,c_2,c_3>$ are given then volume of the parallelopiped is

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

$a\cdot (b\times c)=\left|\begin{array}{ccc}a_1&a_2&a_3\\b_1&b_2&b_3\\c_1&c_2&c_3\end{array}\right|$

Using the above formula we get

$A\cdot (B\times C)=\left|\begin{array}{ccc}-6&3&2\\-2&10&-5\\1&10&-3\end{array}\right|$

Expand along row 1.

$A\cdot (B\times C)=|-6\cdot \det \begin{pmatrix}10&-5\\ 10&-3\end{pmatrix}-3\cdot \det \begin{pmatrix}-2&-5\\ 1&-3\end{pmatrix}+2\cdot \det \begin{pmatrix}-2&10\\ 1&10\end{pmatrix}|$

$A\cdot (B\times C)=-6\cdot (20)-3\cdot (11)+2\cdot (-30)$

$A\cdot (B\times C)=-213$

Volume of the parallelopiped is

$V=|A\cdot (B\times C)|$

$V=|-213|$

$V=213$

Therefore, the volume of the parallelopiped is 213 cubic units.