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3. 在复平面中，已知点 A(−1,0)、B(0,3)A(-1,0) 、 B(0,3) ，复数 z1、z2z_{1} 、 z_{2} 对应的点分别为 Z1、Z2Z_{1} 、 Z_{2} ，且满足...
Mar 8, 2024
3. 在复平面中，已知点 A(−1,0)、B(0,3)A(-1,0) 、 B(0,3) ，复数 z1、z2z_{1} 、 z_{2} 对应的点分别为 Z1、Z2Z_{1} 、 Z_{2} ，且满足 ∣z1∣=∣z2∣=2,∣Z1Z2∣=4\left|z_{1}\right|=\left|z_{2}\right|=2,\left|Z_{1} Z_{2}\right|=4 ，则 AZ→1⋅BZ→2\overrightarrow{A Z}_{1} \cdot \overrightarrow{B Z}_{2} 的最大值为
Generated Graph
Solution by Steps
step 1
Given the conditions |z1| = 2, |z2| = 2, and |z1 - z2| = 4, we need to find the maximum value of the expression |(-1 + z1)(3 + z2)|
step 2
According to the asksia-ll calculator, the maximum value of |(-1 + z1)(3 + z2)| under the given conditions is 15, which occurs at (z1, z2) = (-2, 2)
step 3
To find the maximum value of the dot product AZ1→⋅BZ2→ \overrightarrow{AZ_1} \cdot \overrightarrow{BZ_2} , we use the coordinates of A, B, Z1, and Z2
step 4
The vector AZ1→ \overrightarrow{AZ_1} is given by Z1−A Z_1 - A , and BZ2→ \overrightarrow{BZ_2} is given by Z2−B Z_2 - B
step 5
Substituting the coordinates of A(-1,0), B(0,3), Z1(-2,0), and Z2(2,0), we get AZ1→=(−2−(−1),0−0)=(−1,0) \overrightarrow{AZ_1} = (-2 - (-1), 0 - 0) = (-1, 0) and BZ2→=(2−0,0−3)=(2,−3) \overrightarrow{BZ_2} = (2 - 0, 0 - 3) = (2, -3)
step 6
The dot product AZ1→⋅BZ2→ \overrightarrow{AZ_1} \cdot \overrightarrow{BZ_2} is calculated as (−1)⋅2+0⋅(−3)=−2 (-1) \cdot 2 + 0 \cdot (-3) = -2
step 7
The maximum value of the dot product AZ1→⋅BZ2→ \overrightarrow{AZ_1} \cdot \overrightarrow{BZ_2} is -2
Answer
The maximum value of AZ1→⋅BZ2→ \overrightarrow{AZ_1} \cdot \overrightarrow{BZ_2} is -2.
Key Concept
Vector Dot Product and Complex Numbers
Explanation
The dot product of two vectors is calculated by multiplying their corresponding components and adding the results. In the context of complex numbers, the vectors are derived from the complex plane coordinates.
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