Lewis Moran
12/03/2023 · High School
Solve by viing midpoint method \( \left(\sqrt{\frac{1}{2}}, 3 \sqrt{5}\right)(\sqrt{2}-2 \sqrt{5}) \)
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Tutor-Verified Answer
Step-by-step Solution
To solve the expression \( \left(\sqrt{\frac{1}{2}}, 3 \sqrt{5}\right)(\sqrt{2}-2 \sqrt{5}) \) using the midpoint method, we first need to clarify what the expression means. It seems to involve a vector and a scalar multiplication.
Let's break it down step by step:
1. **Identify the components**: The expression appears to be a vector \( \mathbf{v} = \left(\sqrt{\frac{1}{2}}, 3 \sqrt{5}\right) \) and a scalar \( k = \sqrt{2} - 2\sqrt{5} \).
2. **Scalar multiplication**: To multiply the vector by the scalar, we multiply each component of the vector by the scalar:
\[
\mathbf{v} \cdot k = \left(\sqrt{\frac{1}{2}} \cdot k, 3\sqrt{5} \cdot k\right)
\]
3. **Calculate the scalar**: First, we need to compute \( k = \sqrt{2} - 2\sqrt{5} \).
4. **Multiply the components**:
- For the first component:
\[
\sqrt{\frac{1}{2}} \cdot k = \sqrt{\frac{1}{2}} \cdot (\sqrt{2} - 2\sqrt{5}) = \sqrt{\frac{1}{2}} \cdot \sqrt{2} - \sqrt{\frac{1}{2}} \cdot 2\sqrt{5}
\]
\[
= 1 - 2\sqrt{10}
\]
- For the second component:
\[
3\sqrt{5} \cdot k = 3\sqrt{5} \cdot (\sqrt{2} - 2\sqrt{5}) = 3\sqrt{5} \cdot \sqrt{2} - 3\sqrt{5} \cdot 2\sqrt{5}
\]
\[
= 3\sqrt{10} - 30
\]
5. **Combine the results**: The resulting vector after scalar multiplication is:
\[
\left(1 - 2\sqrt{10}, 3\sqrt{10} - 30\right)
\]
Thus, the final result of the expression \( \left(\sqrt{\frac{1}{2}}, 3 \sqrt{5}\right)(\sqrt{2}-2 \sqrt{5}) \) is:
\[
\left(1 - 2\sqrt{10}, 3\sqrt{10} - 30\right)
\]
Quick Answer
The result is \( \left(1 - 2\sqrt{10}, 3\sqrt{10} - 30\right) \).
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