Suppose `a=5i-3j+2k, b=-i+2j+3k, c=7i-18j+21k," then "[a-b" "b-c" "c-a]=`_______

Answer

To solve the problem, we need to find the scalar triple product of the vectors \( a-b \), \( b-c \), and \( c-a \). Let's go through the steps systematically.

### Step 1: Calculate the vectors \( a-b \), \( b-c \), and \( c-a \)

Given:
- \( a = 5i - 3j + 2k \)
- \( b = -i + 2j + 3k \)
- \( c = 7i - 18j + 21k \)

1. **Calculate \( a - b \)**:
\[ a - b = (5i - 3j + 2k) - (-i + 2j + 3k) = 5i - 3j + 2k + i - 2j - 3k = (5 + 1)i + (-3 - 2)j + (2 - 3)k = 6i - 5j - k \]

2. **Calculate \( b - c \)**:
\[ b - c = (-i + 2j + 3k) - (7i - 18j + 21k) = -i + 2j + 3k - 7i + 18j - 21k = (-1 - 7)i + (2 + 18)j + (3 - 21)k = -8i + 20j - 18k \]

3. **Calculate \( c - a \)**:
\[ c - a = (7i - 18j + 21k) - (5i - 3j + 2k) = 7i - 18j + 21k - 5i + 3j - 2k = (7 - 5)i + (-18 + 3)j + (21 - 2)k = 2i - 15j + 19k \]

### Step 2: Form the matrix using the vectors

Now we can form the matrix using the vectors \( a-b \), \( b-c \), and \( c-a \):
\[ \begin{vmatrix} 6 & -5 & -1 \\ -8 & 20 & -18 \\ 2 & -15 & 19 \end{vmatrix} \]

### Step 3: Calculate the determinant

We can calculate the determinant using the formula for a 3x3 matrix:
\[ \text{Det} = a(ei - fh) - b(di - fg) + c(dh - eg) \]
Where the matrix is:
\[ \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} \]

For our matrix:
- \( a = 6, b = -5, c = -1 \)
- \( d = -8, e = 20, f = -18 \)
- \( g = 2, h = -15, i = 19 \)

Calculating the determinant:
\[ \text{Det} = 6(20 \cdot 19 - (-18)(-15)) - (-5)(-8 \cdot 19 - (-18)(2)) + (-1)(-8 \cdot -15 - 20 \cdot 2) \]

Calculating each term:
1. \( 20 \cdot 19 = 380 \)
2. \( -18 \cdot -15 = 270 \)
3. \( -8 \cdot 19 = -152 \)
4. \( -18 \cdot 2 = -36 \)
5. \( -8 \cdot -15 = 120 \)
6. \( 20 \cdot 2 = 40 \)

Now substituting back:
\[ \text{Det} = 6(380 - 270) - (-5)(-152 + 36) + (-1)(120 - 40) \]
\[ = 6(110) - (-5)(-116) - (80) \]
\[ = 660 - 580 - 80 \]
\[ = 660 - 660 = 0 \]

### Final Answer
The value of the scalar triple product \([a-b, b-c, c-a] = 0\).

---