Let vectors A = (2, 1, -4), B = (-3, 0, 1), and C = (-1, -1, 2). Calculate the following: 2(B · C). Which of the following can be computed? (options: A · B · C, A · (B · C), A · (B... Let vectors A = (2, 1, -4), B = (-3, 0, 1), and C = (-1, -1, 2). Calculate the following: 2(B · C). Which of the following can be computed? (options: A · B · C, A · (B · C), A · (B + C), 3 · A). Express your answer in terms of V1.
Understand the Problem
The question is asking to calculate specific vector operations involving the given vectors A, B, and C. The user must compute scalar products and identify which operations can be performed with the vectors. This involves understanding vector mathematics, specifically dot products and scalar multiplications.
Answer
The answer is \( 10 \).
Answer for screen readers
The value for ( 2(\mathbf{B} \cdot \mathbf{C}) ) is ( 10 ).
Steps to Solve
- Calculate the dot product ( \mathbf{B} \cdot \mathbf{C} )
First, compute the dot product of vectors ( \mathbf{B} ) and ( \mathbf{C} ):
[ \mathbf{B} = (-3, 0, 1) ] [ \mathbf{C} = (-1, -1, 2) ]
The dot product is given by:
$$ \mathbf{B} \cdot \mathbf{C} = (-3)(-1) + (0)(-1) + (1)(2) $$
Now calculate:
$$ \mathbf{B} \cdot \mathbf{C} = 3 + 0 + 2 = 5 $$
- Multiply the result by 2
Now, multiply the result from the previous step by 2:
$$ 2(\mathbf{B} \cdot \mathbf{C}) = 2(5) = 10 $$
- Identify computable operations
To check which vector operations can be computed, consider:
- ( \mathbf{A} \cdot \mathbf{B} )
- ( \mathbf{A} \cdot (\mathbf{B} \cdot \mathbf{C}) )
- ( \mathbf{A} \cdot (\mathbf{B} + \mathbf{C}) )
- ( 3\mathbf{A} )
Since the dot product operation can only be performed between vectors and not scalar results, you can compute:
- ( \mathbf{A} \cdot \mathbf{B} )
- ( 3\mathbf{A} )
The operations involving the dot product between ( \mathbf{A} ) and the result of another dot product will not work.
The value for ( 2(\mathbf{B} \cdot \mathbf{C}) ) is ( 10 ).
More Information
The result ( 10 ) comes from computing the dot product of vectors ( \mathbf{B} ) and ( \mathbf{C} ), followed by multiplying that result by 2.
Tips
- Forgetting to apply proper order of operations when computing dot products.
- Confusing the scalar result of a dot product with the vectors themselves, which can lead to incorrect assumptions about computable operations.
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