Calculate 2(B ⋅ 3C)
Understand the Problem
The question is asking to calculate the expression involving the dot product of two vectors, specifically 2 times the dot product of vector B and 3 times vector C.
Answer
$30$
Answer for screen readers
The final answer is $30$.
Steps to Solve
- Calculate the scalar multiplication of vector C
First, we find ( 3C ), where ( C = (-1, -1, 2) ).
[ 3C = 3 \cdot (-1, -1, 2) = (-3, -3, 6) ]
- Calculate the dot product of vector B and 3C
Now, we compute the dot product ( B \cdot (3C) ) with ( B = (-3, 0, 1) ).
The dot product formula is given by: [ B \cdot (3C) = B_1(3C_1) + B_2(3C_2) + B_3(3C_3) ] Substituting in the values: [ B \cdot (3C) = (-3)(-3) + (0)(-3) + (1)(6) ] Calculating each term: [ = 9 + 0 + 6 = 15 ]
- Calculate the final expression
Finally, we multiply the dot product result by 2: [ 2(B \cdot (3C)) = 2 \cdot 15 = 30 ]
The final answer is $30$.
More Information
This computation combines scalar multiplication followed by the dot product of two vectors. The dot product provides a measure of how closely two vectors align in direction, and scalar multiplication adjusts their magnitude.
Tips
- Forgetting to multiply vector C by the scalar before calculating the dot product.
- Miscalculating the dot product by omitting one of the components or summing incorrectly.
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