ab-3bc+4cd a(3;-1;1) b(1;3;2) c(1;-1;-1) d(4;0;3)

To evaluate the expression \( ab - 3bc + 4cd \) using the vectors given, we first need to compute the dot products of the vectors involved according to the specified expression.

Given the following vectors:

• \( \mathbf{a} = (3, -1, 1) \)
• \( \mathbf{b} = (1, 3, 2) \)
• \( \mathbf{c} = (1, -1, -1) \)
• \( \mathbf{d} = (4, 0, 3) \)

Step 1: Calculate \( ab \)

The dot product \( ab \) is calculated as follows:

\[ \mathbf{a} \cdot \mathbf{b} = (3)(1) + (-1)(3) + (1)(2) = 3 - 3 + 2 = 2 \]

Step 2: Calculate \( 3bc \)

Next, we calculate \( \mathbf{b} \cdot \mathbf{c} \):

\[ \mathbf{b} \cdot \mathbf{c} = (1)(1) + (3)(-1) + (2)(-1) = 1 - 3 - 2 = -4 \]

Now we calculate \( 3bc \):

\[ 3bc = 3 \times (-4) = -12 \]

Step 3: Calculate \( 4cd \)

Now we calculate \( \mathbf{c} \cdot \mathbf{d} \):

\[ \mathbf{c} \cdot \mathbf{d} = (1)(4) + (-1)(0) + (-1)(3) = 4 + 0 - 3 = 1 \]

Now we calculate \( 4cd \):

\[ 4cd = 4 \times 1 = 4 \]

Step 4: Combine the results

Now we can combine all of these results to find \( ab - 3bc + 4cd \):

\[ ab - 3bc + 4cd = 2 - (-12) + 4 \]

This simplifies to:

\[ = 2 + 12 + 4 = 18 \]

Thus, the final result is:

\[ \boxed{18} \]