If `vec(F ) = hat(i) +2 hat(j) + hat(k) and vec(V) = 4hat(i) - hat(j) + 7hat(k)` what is `vec(F) . vec(v)` ?

Explanation

To find the dot product of the vectors \(\vec{F}\) and \(\vec{V}\), we can follow these steps:

### Step 1: Write down the vectors
Given:
\[ \vec{F} = \hat{i} + 2\hat{j} + \hat{k} \]
\[ \vec{V} = 4\hat{i} - \hat{j} + 7\hat{k} \]

### Step 2: Apply the dot product formula
The dot product of two vectors \(\vec{A} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}\) and \(\vec{B} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}\) is given by:
\[ \vec{A} \cdot \vec{B} = a_1b_1 + a_2b_2 + a_3b_3 \]

### Step 3: Identify the components
From the vectors:
- For \(\vec{F}\): \(a_1 = 1\), \(a_2 = 2\), \(a_3 = 1\)
- For \(\vec{V}\): \(b_1 = 4\), \(b_2 = -1\), \(b_3 = 7\)

### Step 4: Calculate the dot product
Now, substitute the components into the dot product formula:
\[ \vec{F} \cdot \vec{V} = (1)(4) + (2)(-1) + (1)(7) \]

### Step 5: Perform the calculations
Calculating each term:
- First term: \(1 \cdot 4 = 4\)
- Second term: \(2 \cdot -1 = -2\)
- Third term: \(1 \cdot 7 = 7\)

Now, add these results together:
\[ \vec{F} \cdot \vec{V} = 4 - 2 + 7 \]
\[ = 2 + 7 = 9 \]

### Final Answer
Thus, the dot product \(\vec{F} \cdot \vec{V} = 9\).

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