Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS. P(-2, 1, 0), Q(2, 3, 2), R(1, 4, -1), S(3, 6, 1)

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Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS.  P(-2, 1, 0), Q(2, 3, 2), R(1, 4, -1), S(3, 6, 1)

pierretteA · Accepted Answer

Any three vectors&amp;nbsp;(a=,b=)&amp;nbsp;and&amp;nbsp;(c=,)&amp;nbsp;we can find the volume of the parallelogram that the three vectors are shaping, using the triple product of these three vectors&amp;nbsp;a⋅(b×c)Volume=[a⋅(b×c)]&amp;nbsp;(1)[a⋅(b×c)]=|a1a2a3b1b2b3c1c2c3|The value&amp;nbsp;△&amp;nbsp;of the determined can be computed,|a1a2a3b1b2b3c1c2c3|=a1|b2b3c2c3|−a2|b1b3c1c3|+a3|b1b2c1c2|△=a1((b2c3)−(b3c2))−a2((b1c3)−(b3c1))+a3((b1c2)−(b2c1))Or the value of the determinant can be found using Sarrus&#039;s rule&amp;nbsp;3×3&amp;nbsp;determinant &quot;This work for&amp;nbsp;3×3&amp;nbsp;determinant Only&quot;, where by duplicating the first two columuns of the matrix after the third columns, a&amp;nbsp;3×5&amp;nbsp;determinant is formed as follows|a1a2a3b1b2b3c1c2c3||a1a2b1b2c1c2|The value of the determinant is thus determined by Sarrus&#039;s rule to be the product of all the South-East diagonals minus the product of all the South-West diagonals, so that△=((a1b2c3)+(a2b3c1)+(a3b1c2))−((a2b1c3)+(a1b3c2)+(a3b2c1))In order to get the magnitude for the triple product, which would reflect the parallelogram&#039;s volume, we start by determining the value of the triple product, also known as &quot;the value of the determinant.&quot;PQ=Q−P=⟨2+2,3−1,2−0⟩=⟨4,2,2⟩PS=S−P=⟨3+2,6−1,1−0⟩=⟨5,5,1⟩PR=R−P=⟨1+2,4−1,−1−0⟩=⟨3,3,−1⟩PS⋅(PQ×PR)=|55142233−1|△=|55

nick1337 · Accepted Answer

The scalar triple product is defined as follows:PQ→·(PR→×PS→)where · denotes the dot product and × denotes the cross product.First, let&#039;s find the vectors PQ→, PR→, and PS→.PQ→=[2−(−2)3−12−0]=[422]PR→=[1−(−2)4−1−1−0]=[33−1]PS→=[3−(−2)6−11−0]=[551]Next, let&#039;s calculate the cross product PR→×PS→.PR→×PS→=[33−1]×[551]Using the cross product formula, we can compute the components of the resulting vector:[33−1]×[551]=[(3×1)−(3×5)(3×5)−(3×1)(3×5)−(3×5)]=[−12120]Now, we can calculate the dot product of PQ→ and PR→×PS→.PQ→·(PR→×PS→)=[422]·[−12120]Using the dot product formula, we have:[422]·[−12120]=(4×−12)+(2×12)+(2×0)=−48+24+0=−24Therefore, the volume of the parallelepiped with adjacent edges PQ→, PR→, and PS→ is |−24|=24 cubic units.

Vasquez · Accepted Answer

To find the volume of a parallelepiped with adjacent edges PQ, PR, and PS, we can use the scalar triple product. The scalar triple product of three vectors is defined as follows:𝐕=PQ·(PR×PS)where · denotes the dot product and × denotes the cross product.First, let&#039;s find the vectors PQ, PR, and PS. The vector PQ is given by:PQ=𝐐−𝐏=(2,3,2)−(−2,1,0)=(4,2,2)Similarly, we can find the vectors PR and PS:PR=𝐑−𝐏=(1,4,−1)−(−2,1,0)=(3,3,−1)PS=𝐒−𝐏=(3,6,1)−(−2,1,0)=(5,5,1)Next, we can calculate the cross product of PR and PS:PR×PS=|𝐢𝐣𝐤33−1551|Expanding the determinant, we get:PR×PS=(3·1−(−1)·5)𝐢−(3·1−(−1)·5)𝐣+(3·5−5·5)𝐤=8𝐢−8𝐣−10𝐤=(8,−8,−10)Now, we can calculate the dot product of PQ and (PR×PS):PQ·(PR×PS)=(4,2,2)·(8,−8,−10)=4·8+2·(−8)+2·(−10)=32−16−20=−4Finally, we take the absolute value of the scalar triple product to get the volume of the parallelepiped:Volume=|𝐕|=|−4|=4Therefore, the volume of the parallelepiped with adjacent edges PQ, PR, and PS is 4.

RizerMix · Accepted Answer

Result:72 cubic unitsSolution:|PQ·(PR×PS)|where PQ is the vector from point P to Q, and PR×PS is the cross product of vectors PR and PS. Let&#039;s calculate each component step by step.First, let&#039;s find the vector PQ:PQ=[2−(−2)3−12−0]=[422]Next, let&#039;s calculate the cross product PR×PS:PR×PS=[1−(−2)4−1−1−0]×[3−(−2)6−11−0]=[33−1]×[551]=[(3×1)−(3×5)(3×1)−(3×5)(3×5)−(3×5)]=[−12−120]Now, let&#039;s calculate the scalar triple product:|PQ·(PR×PS)|=|[422]·[−12−120]|=|(4×−12)+(2×−12)+(2×0)|=|−48−24+0|=|−72|=72Therefore, the volume of the parallelepiped is 72 cubic units.

Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS. P(-2, 1, 0), Q(2, 3, 2), R(1, 4, -1), S(3, 6, 1)

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