If `a=2i-3j+5k, b=3i-4j+5k" and "c=5i-3j-2k` then volume of the parallelopiped with coterminus edges `a+b, b+c, c+a` is _______

Answer

To find the volume of the parallelepiped with coterminous edges \( \mathbf{a} + \mathbf{b}, \mathbf{b} + \mathbf{c}, \mathbf{c} + \mathbf{a} \), we will follow these steps:

### Step 1: Calculate the vectors \( \mathbf{a} + \mathbf{b}, \mathbf{b} + \mathbf{c}, \mathbf{c} + \mathbf{a} \)

Given:
- \( \mathbf{a} = 2\mathbf{i} - 3\mathbf{j} + 5\mathbf{k} \)
- \( \mathbf{b} = 3\mathbf{i} - 4\mathbf{j} + 5\mathbf{k} \)
- \( \mathbf{c} = 5\mathbf{i} - 3\mathbf{j} - 2\mathbf{k} \)

1. **Calculate \( \mathbf{a} + \mathbf{b} \)**:
\[ \mathbf{a} + \mathbf{b} = (2 + 3)\mathbf{i} + (-3 - 4)\mathbf{j} + (5 + 5)\mathbf{k} = 5\mathbf{i} - 7\mathbf{j} + 10\mathbf{k} \]

2. **Calculate \( \mathbf{b} + \mathbf{c} \)**:
\[ \mathbf{b} + \mathbf{c} = (3 + 5)\mathbf{i} + (-4 - 3)\mathbf{j} + (5 - 2)\mathbf{k} = 8\mathbf{i} - 7\mathbf{j} + 3\mathbf{k} \]

3. **Calculate \( \mathbf{c} + \mathbf{a} \)**:
\[ \mathbf{c} + \mathbf{a} = (5 + 2)\mathbf{i} + (-3 - 3)\mathbf{j} + (-2 + 5)\mathbf{k} = 7\mathbf{i} - 6\mathbf{j} + 3\mathbf{k} \]

### Step 2: Set up the matrix for the scalar triple product

The volume \( V \) of the parallelepiped can be calculated using the scalar triple product:
\[ V = |\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})| \]
where \( \mathbf{u} = \mathbf{a} + \mathbf{b} \), \( \mathbf{v} = \mathbf{b} + \mathbf{c} \), and \( \mathbf{w} = \mathbf{c} + \mathbf{a} \).

The vectors are:
- \( \mathbf{u} = 5\mathbf{i} - 7\mathbf{j} + 10\mathbf{k} \)
- \( \mathbf{v} = 8\mathbf{i} - 7\mathbf{j} + 3\mathbf{k} \)
- \( \mathbf{w} = 7\mathbf{i} - 6\mathbf{j} + 3\mathbf{k} \)

### Step 3: Form the determinant

The volume can be calculated using the determinant:
\[ V = \begin{vmatrix} 5 & -7 & 10 \\ 8 & -7 & 3 \\ 7 & -6 & 3 \end{vmatrix} \]

### Step 4: Calculate the determinant

Using the determinant formula:
\[ \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg) \]

We calculate:
\[ = 5((-7)(3) - (-6)(3)) - (-7)(8(3) - 7(3)) + 10(8(-6) - (-7)(7)) \]
Calculating step-by-step:
1. \( = 5(-21 + 18) \)
2. \( = 5(-3) = -15 \)

Next:
1. \( -(-7)(24 - 21) = 7 \times 3 = 21 \)
2. \( 10(-48 + 49) = 10(1) = 10 \)

Now combine:
\[ = -15 + 21 + 10 = 16 \]

### Step 5: Final volume

Thus, the volume of the parallelepiped is:
\[ V = |16| = 16 \text{ cubic units} \]