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}
]
