Find the dot product of these two vectors. $$ \begin{array}{l}\overrightarrow{\mathrm{v}}=\langle-1,4,5\rangle \ \overrightarrow{\mathrm{w}}=\langle 1,-4,5\rangle \ \overrightarrow{\mathrm{v}} \cdot \overrightarrow{\mathrm{w}}=[?]\end{array} $$

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To find the dot product of the vectors $$\overrightarrow{\mathrm{v}}$$ and $$\overrightarrow{\mathrm{w}}$$, you can use the following formula:

$$
\overrightarrow{\mathrm{v}} \cdot \overrightarrow{\mathrm{w}} = v_1w_1 + v_2w_2 + v_3w_3
$$

Given:
$$
\overrightarrow{\mathrm{v}} = \langle -1, 4, 5 \rangle
$$
$$
\overrightarrow{\mathrm{w}} = \langle 1, -4, 5 \rangle
$$

Let's compute each component:

1. **First Components:**
   $$
   v_1w_1 = (-1) \times 1 = -1
   $$

2. **Second Components:**
   $$
   v_2w_2 = 4 \times (-4) = -16
   $$

3. **Third Components:**
   $$
   v_3w_3 = 5 \times 5 = 25
   $$

Now, sum these results to find the dot product:

$$
\overrightarrow{\mathrm{v}} \cdot \overrightarrow{\mathrm{w}} = (-1) + (-16) + 25 = 8
$$

**Answer:**  
$$
\overrightarrow{\mathrm{v}} \cdot \overrightarrow{\mathrm{w}} = 8
$$
The dot product of the vectors is 8.

Find the dot product of these two vectors. \[ \begin{array}{l}\overrightarrow{\mathrm{v}}=\langle-1,4,5\rangle \\ \overrightarrow{\mathrm{w}}=\langle 1,-4,5\rangle \\ \overrightarrow{\mathrm{v}} \cdot \overrightarrow{\mathrm{w}}=[?]\end{array} \]

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