Find the volume of the parallelepiped determined by the vectors a, b, and c. a=&lt;1, 2, 3&gt;, b=&lt;-1, 1, 2&gt;, c=&lt;2, 1, 4&gt;

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Find the volume of the parallelepiped determined by the vectors a, b, and c. a=&amp;lt;1, 2, 3&amp;gt;, b=&amp;lt;-1, 1, 2&amp;gt;, c=&amp;lt;2, 1, 4&amp;gt;

Sadie Eaton · Accepted Answer

Calculations:&amp;nbsp;We compute the triple product of the three given vectors to find the volume of the parallelogram they are creating. The triple product of the three vectors can be computed for the vectors in any different order. We begin by assigning the vectors in the determinant as follows.c⋅(a×b)=|214123−112|&amp;nbsp;△=|214123−112||2112−11|&amp;nbsp;We then, calculate the value&amp;nbsp;△&amp;nbsp;of the determinant, using Leibniz or Sarrus, using Sarruss.

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Question: Find the volume of the parallelepiped determined by the vectors a,b and c.  a= 1, 2,3 , b= 1,1,2 and c= 1,1,2

user_27qwe · Accepted Answer

Step 1:The scalar triple product of three vectors is defined as:V=|a·(b×c)| where a·(b×c) represents the dot product between a and the cross product of b and c. The magnitude of the scalar triple product gives us the volume of the parallelepiped.Let&#039;s calculate the dot product a·(b×c) step by step:First, we find the cross product of b and c:b×c=|𝐢𝐣𝐤−112214|Expanding the determinant, we get:b×c=(1·4−2·1)𝐢−(−1·4−2·2)𝐣+(−1·1−2·1)𝐤Simplifying further, we have:b×c=2𝐢−(−8)𝐣−3𝐤=2𝐢+8𝐣−3𝐤Step 2:Next, we calculate the dot product of a and the resulting vector (b×c):a·(b×c)=(123)·(28−3)Expanding the dot product, we get:a·(b×c)=(1·2)+(2·8)+(3·−3)=2+16−9=9Finally, we take the absolute value of the scalar triple product to obtain the volume of the parallelepiped:V=|9|=9Therefore, the volume of the parallelepiped determined by the vectors 𝐚=(123), 𝐛=(−112), and 𝐜=(214) is 9.

karton · Accepted Answer

Answer:13Solution:Volume=|𝐚·(𝐛×𝐜)|.First, we need to compute the cross product of 𝐛 and 𝐜:𝐛×𝐜=(−112)×(214)=(1−83).Next, we calculate the dot product of 𝐚 with 𝐛×𝐜:𝐚·(𝐛×𝐜)=(123)·(1−83)=1·1+2·(−8)+3·3=−13.Finally, we take the absolute value of the dot product to obtain the volume:Volume=|𝐚·(𝐛×𝐜)|=|−13|=13.

star233 · Accepted Answer

To find the volume of the parallelepiped determined by the vectors 𝐚=⟨1,2,3⟩, 𝐛=⟨−1,1,2⟩, and 𝐜=⟨2,1,4⟩, we can use the scalar triple product. The volume V is given by the absolute value of the scalar triple product of the vectors 𝐚, 𝐛, and 𝐜.The scalar triple product is defined as follows:(𝐚×𝐛)·𝐜=|a1a2a3b1b2b3c1c2c3|Let&#039;s calculate the scalar triple product:(𝐚×𝐛)·𝐜&amp;amp;=|123−112214|&amp;amp;=(1·1·4)+(2·2·2)+(3·(−1)·1)−(3·1·2)−(2·(−1)·4)−(1·2·1)&amp;amp;=4+8−3−6−8−2&amp;amp;=−7The volume V is the absolute value of the scalar triple product:V=|𝐚·(𝐛×𝐜)|=|−7|=7Therefore, the volume of the parallelepiped determined by the vectors 𝐚=⟨1,2,3⟩, 𝐛=⟨−1,1,2⟩, and 𝐜=⟨2,1,4⟩ is 7.

Find the volume of the parallelepiped determined by the vectors a, b, and c. a=<1, 2, 3>, b=<-1, 1, 2>, c=<2, 1, 4>

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