Let vectors A = (2, 1, -4), B = (-3, 0, 1), and C = (-1, -1, 2). Calculate: 2(B · C). Which of the following can be computed?
Understand the Problem
The question involves calculations with vectors A, B, and C. It's asking to compute the scalar value of the expression '2(B · C)' and to determine which of the provided expressions can also be computed based on the vectors.
Answer
$10$
Answer for screen readers
The value of ( 2(B \cdot C) ) is ( 10 ).
Steps to Solve
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Calculate the dot product ( B \cdot C )
The dot product of two vectors ( B ) and ( C ) is calculated using the formula:
$$ B \cdot C = B_x \cdot C_x + B_y \cdot C_y + B_z \cdot C_z $$
For vectors ( B = (-3, 0, 1) ) and ( C = (-1, -1, 2) ):
$$ B \cdot C = (-3)(-1) + (0)(-1) + (1)(2) = 3 + 0 + 2 = 5 $$
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Calculate ( 2(B \cdot C) )
Now, we multiply the result of the dot product by 2:
$$ 2(B \cdot C) = 2 \cdot 5 = 10 $$
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Determine which expressions can be computed
Review the provided options:
- ( A \cdot (B \cdot C) ) - Not valid since that's a scalar.
- ( A \cdot (B \cdot C) ) - Same reasoning as above.
- ( A \cdot (B + C) ) - Valid, as ( B + C ) results in a vector.
- ( 3 \cdot A ) - Valid, as scalar multiplication is defined.
The value of ( 2(B \cdot C) ) is ( 10 ).
More Information
Dot products yield scalar values, and operations on vectors can include addition and scalar multiplication. The value of ( 2(B \cdot C) ) demonstrates the relationship between the vectors through the dot product. The correct computations remain consistent with vector algebra rules.
Tips
- Confusing dot products with scalar products: the dot product results in a scalar, not a vector.
- Misapplying vector addition or scalar multiplication rules: always ensure to add vectors before attempting further operations.
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