If a, b and c are the three vectors mutually perpendicular to each other to form a right handed system and `|a|=1, |b|=3 and |c|=5`, then `[a-2b b-3c c-4a]` is equal to

Options

A. 0
B. `-24`
C. 3600
D. `-215`

Explanation

To solve the problem, we need to find the scalar triple product of the vectors \( \mathbf{A} - 2\mathbf{B} \), \( \mathbf{B} - 3\mathbf{C} \), and \( \mathbf{C} - 4\mathbf{A} \). Let's denote these vectors as follows:

- \( \mathbf{u} = \mathbf{A} - 2\mathbf{B} \)
- \( \mathbf{v} = \mathbf{B} - 3\mathbf{C} \)
- \( \mathbf{w} = \mathbf{C} - 4\mathbf{A} \)

### Step 1: Express the vectors in terms of their components
Given that \( |\mathbf{A}| = 1 \), \( |\mathbf{B}| = 3 \), and \( |\mathbf{C}| = 5 \), we can represent these vectors in a right-handed coordinate system as:
- \( \mathbf{A} = (1, 0, 0) \)
- \( \mathbf{B} = (0, 3, 0) \)
- \( \mathbf{C} = (0, 0, 5) \)

### Step 2: Calculate \( \mathbf{u} \), \( \mathbf{v} \), and \( \mathbf{w} \)
Now, we can calculate each vector:
- \( \mathbf{u} = \mathbf{A} - 2\mathbf{B} = (1, 0, 0) - 2(0, 3, 0) = (1, -6, 0) \)
- \( \mathbf{v} = \mathbf{B} - 3\mathbf{C} = (0, 3, 0) - 3(0, 0, 5) = (0, 3, -15) \)
- \( \mathbf{w} = \mathbf{C} - 4\mathbf{A} = (0, 0, 5) - 4(1, 0, 0) = (-4, 0, 5) \)

### Step 3: Compute the scalar triple product \( [\mathbf{u}, \mathbf{v}, \mathbf{w}] \)
The scalar triple product can be calculated using the determinant of a matrix formed by the vectors:
\[ [\mathbf{u}, \mathbf{v}, \mathbf{w}] = \begin{vmatrix} 1 & -6 & 0 \\ 0 & 3 & -15 \\ -4 & 0 & 5 \end{vmatrix} \]

### Step 4: Calculate the determinant
To compute the determinant, we can use the formula for a 3x3 matrix:
\[ \text{det} = a(ei-fh) - b(di-fg) + c(dh-eg) \]
Substituting the values:
\[ = 1(3 \cdot 5 - (-15) \cdot 0) - (-6)(0 \cdot 5 - (-15)(-4)) + 0(0 \cdot 0 - 3 \cdot (-4)) \]
\[ = 1(15) + 6(60) + 0 \]
\[ = 15 + 360 = 375 \]

### Final Answer
Thus, the scalar triple product \( [\mathbf{A} - 2\mathbf{B}, \mathbf{B} - 3\mathbf{C}, \mathbf{C} - 4\mathbf{A}] \) is equal to \( 375 \).