Question #178942

A parallelepiped is defined by the following vectors |𝑎| = (2, 3, −1), |𝑏| = (−1, 0, 2),

and |𝑐| = (3, −1, 2). Find the volume of the parallelepiped. 


1
Expert's answer
2021-05-12T05:04:23-0400

The volume is given by the formula ab×c.|a\cdot |b\times c||. We can write b=i^+2k^b= -\hat{i}+2\hat{k} and c=3i^j^+2k^.c=3\hat{i}-\hat{j}+2\hat{k}. b×c=i^(2×1)+j^(2×3(1)×2)+k^(1×1)b\times c=\hat{i}(-2\times -1)+\hat{j}(2\times 3-(-1)\times 2)+\hat{k}(-1\times -1) =2i^+8j^+k^.=2\hat{i}+8\hat{j}+\hat{k}. Hence a(b×c)=a\cdot(b\times c)= (2i^+3j^k^)(2i^+8j^+k^)(2\hat{i}+3\hat{j}-\hat{k})\cdot (2\hat{i}+8\hat{j}+\hat{k}) = 2×(2)+(3)×8(1×1)=27.2\times(2)+(3)\times 8-(1\times 1)=27. Hence volume =27=27|27|=27 in cube units.


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