The vector `((i-j) times (j-k)) times (i+5k)` is equal to

Options

A. `5i-4j-k`
B. `3i-2j+5k`
C. `4i-5j-k`
D. `5i+4j-k`

Answer

To solve the problem, we need to evaluate the expression \(((\mathbf{i} - \mathbf{j}) \times (\mathbf{j} - \mathbf{k})) \times (\mathbf{i} + 5\mathbf{k})\).

### Step 1: Find the first cross product \((\mathbf{i} - \mathbf{j}) \times (\mathbf{j} - \mathbf{k})\)

Let's denote:
- \(\mathbf{a} = \mathbf{i} - \mathbf{j}\)
- \(\mathbf{b} = \mathbf{j} - \mathbf{k}\)

Now we can compute the cross product \(\mathbf{a} \times \mathbf{b}\):
\[ \mathbf{a} \times \mathbf{b} = (\mathbf{i} - \mathbf{j}) \times (\mathbf{j} - \mathbf{k}) \]

Using the distributive property of the cross product:
\[ \mathbf{a} \times \mathbf{b} = \mathbf{i} \times \mathbf{j} - \mathbf{i} \times \mathbf{k} - \mathbf{j} \times \mathbf{j} + \mathbf{j} \times \mathbf{k} \]

We know that:
- \(\mathbf{i} \times \mathbf{j} = \mathbf{k}\)
- \(\mathbf{i} \times \mathbf{k} = -\mathbf{j}\)
- \(\mathbf{j} \times \mathbf{j} = \mathbf{0}\)
- \(\mathbf{j} \times \mathbf{k} = \mathbf{i}\)

Thus, substituting these values:
\[ \mathbf{a} \times \mathbf{b} = \mathbf{k} - (-\mathbf{j}) + \mathbf{i} = \mathbf{k} + \mathbf{j} + \mathbf{i} \]
So, we have:
\[ \mathbf{a} \times \mathbf{b} = \mathbf{i} + \mathbf{j} + \mathbf{k} \]

### Step 2: Now compute the second cross product \((\mathbf{i} + \mathbf{j} + \mathbf{k}) \times (\mathbf{i} + 5\mathbf{k})\)

Let:
- \(\mathbf{c} = \mathbf{i} + 5\mathbf{k}\)

Now we compute:
\[ (\mathbf{i} + \mathbf{j} + \mathbf{k}) \times (\mathbf{i} + 5\mathbf{k}) \]

Using the distributive property again:
\[ = \mathbf{i} \times \mathbf{i} + \mathbf{i} \times 5\mathbf{k} + \mathbf{j} \times \mathbf{i} + \mathbf{j} \times 5\mathbf{k} + \mathbf{k} \times \mathbf{i} + \mathbf{k} \times 5\mathbf{k} \]

We know:
- \(\mathbf{i} \times \mathbf{i} = \mathbf{0}\)
- \(\mathbf{i} \times \mathbf{k} = -\mathbf{j}\)
- \(\mathbf{j} \times \mathbf{i} = \mathbf{k}\)
- \(\mathbf{j} \times \mathbf{k} = \mathbf{i}\)
- \(\mathbf{k} \times \mathbf{i} = \mathbf{j}\)
- \(\mathbf{k} \times \mathbf{k} = \mathbf{0}\)

Substituting these values:
\[ = 0 + 5(-\mathbf{j}) + \mathbf{k} + 5\mathbf{i} + \mathbf{j} + 0 \]
\[ = -5\mathbf{j} + \mathbf{k} + 5\mathbf{i} + \mathbf{j} \]
\[ = 5\mathbf{i} - 4\mathbf{j} + \mathbf{k} \]

### Final Answer
Thus, the final result of the expression \(((\mathbf{i} - \mathbf{j}) \times (\mathbf{j} - \mathbf{k})) \times (\mathbf{i} + 5\mathbf{k})\) is:
\[ \boxed{5\mathbf{i} - 4\mathbf{j} + \mathbf{k}} \]