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If `|(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))| =5`, then the value of
`Delta = |(b_(2) c_(3) - b_(3) c_(2),a_(3) c_(2) - a_(2) c_(3),a_(2) b_(3) -a_(3) b_(2)),(b_(3) c_(1) - b_(1) c_(3),a_(1) c_(3) - a_(3) c_(1),a_(3) b_(1) - a_(1) b_(3)),(b_(1) c_(2) - b_(2) c_(1),a_(2) c_(1) - a_(1) c_(2),a_(1) b_(2) - a_(2) b_(1))|` is

A

5

B

25

C

125

D

0

Text Solution

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The correct Answer is:
To solve the problem, we need to evaluate the determinant \( \Delta \) given that the determinant of the matrix formed by the vectors \((a_1, b_1, c_1)\), \((a_2, b_2, c_2)\), and \((a_3, b_3, c_3)\) is equal to 5. We denote this determinant as \( |(a_1, b_1, c_1), (a_2, b_2, c_2), (a_3, b_3, c_3)| = 5 \). ### Step 1: Understand the given determinant We have: \[ |(a_1, b_1, c_1), (a_2, b_2, c_2), (a_3, b_3, c_3)| = 5 \] This means that the volume of the parallelepiped formed by these vectors is 5. ### Step 2: Write down the expression for \( \Delta \) The expression for \( \Delta \) is given as: \[ \Delta = |(b_2 c_3 - b_3 c_2, a_3 c_2 - a_2 c_3, a_2 b_3 - a_3 b_2), (b_3 c_1 - b_1 c_3, a_1 c_3 - a_3 c_1, a_3 b_1 - a_1 b_3), (b_1 c_2 - b_2 c_1, a_2 c_1 - a_1 c_2, a_1 b_2 - a_2 b_1)| \] ### Step 3: Recognize the structure of the determinant The determinant \( \Delta \) can be interpreted as a determinant of a matrix formed by the cross products of the original vectors. The terms in the determinant represent the components of the cross products of the vectors. ### Step 4: Calculate \( \Delta \) Using the properties of determinants, particularly that the determinant of a matrix formed by cross products of vectors is related to the original determinant, we can express \( \Delta \) in terms of the original determinant. The determinant of the matrix formed by the cross products of the vectors is given by: \[ \Delta = |(b_2 c_3 - b_3 c_2, a_3 c_2 - a_2 c_3, a_2 b_3 - a_3 b_2), (b_3 c_1 - b_1 c_3, a_1 c_3 - a_3 c_1, a_3 b_1 - a_1 b_3), (b_1 c_2 - b_2 c_1, a_2 c_1 - a_1 c_2, a_1 b_2 - a_2 b_1)| \] is equal to \( |A|^2 \), where \( |A| \) is the original determinant. ### Step 5: Substitute the known value Since we know that \( |A| = 5 \): \[ \Delta = |A|^2 = 5^2 = 25 \] ### Final Answer Thus, the value of \( \Delta \) is: \[ \Delta = 25 \]
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