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已知向量 a⃗，b⃗\vec{a} ， \vec{b} ， c⃗\vec{c} 满足: ∣a⃗+b⃗∣=3，∣c⃗∣=1|\vec{a}+\vec{b}|=3 ，|\vec{c}|=1 且 $\...
Mar 8, 2024
已知向量 a⃗，b⃗\vec{a} ， \vec{b} ， c⃗\vec{c} 满足: ∣a⃗+b⃗∣=3，∣c⃗∣=1|\vec{a}+\vec{b}|=3 ，|\vec{c}|=1 且 a⃗⋅b⃗+1=(a⃗+b⃗)⋅c⃗\vec{a} \cdot \vec{b}+1=(\vec{a}+\vec{b}) \cdot \vec{c} ，则 ∣a⃗−b⃗∣|\vec{a}-\vec{b}| 的取值范围是
Solution by Steps
step 1
Use the given information to write the equation involving the dot product: ∣a⃗+b⃗∣=3|\vec{a} + \vec{b}| = 3 and a⃗⋅b⃗+1=(a⃗+b⃗)⋅c⃗\vec{a} \cdot \vec{b} + 1 = (\vec{a} + \vec{b}) \cdot \vec{c}
step 2
Square both sides of the first equation to get (a⃗+b⃗)⋅(a⃗+b⃗)=9(\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b}) = 9
step 3
Expand the dot product to get a⃗⋅a⃗+2a⃗⋅b⃗+b⃗⋅b⃗=9\vec{a} \cdot \vec{a} + 2\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} = 9
step 4
Use the fact that ∣c⃗∣=1|\vec{c}| = 1 and normalize the equation a⃗⋅b⃗+1=(a⃗+b⃗)⋅c⃗\vec{a} \cdot \vec{b} + 1 = (\vec{a} + \vec{b}) \cdot \vec{c} to get a⃗⋅b⃗+1=∣a⃗+b⃗∣∣c⃗∣cos⁡θ\vec{a} \cdot \vec{b} + 1 = |\vec{a} + \vec{b}| |\vec{c}| \cos{\theta}, where θ\theta is the angle between a⃗+b⃗\vec{a} + \vec{b} and c⃗\vec{c}
step 5
Since ∣a⃗+b⃗∣=3|\vec{a} + \vec{b}| = 3 and ∣c⃗∣=1|\vec{c}| = 1, we have a⃗⋅b⃗+1=3cos⁡θ\vec{a} \cdot \vec{b} + 1 = 3 \cos{\theta}
step 6
Substitute a⃗⋅b⃗\vec{a} \cdot \vec{b} from step 5 into the equation from step 3 to get a⃗⋅a⃗+2(3cos⁡θ−1)+b⃗⋅b⃗=9\vec{a} \cdot \vec{a} + 2(3 \cos{\theta} - 1) + \vec{b} \cdot \vec{b} = 9
step 7
Now, consider the absolute value of a⃗−b⃗\vec{a} - \vec{b} and square it to get ∣a⃗−b⃗∣2=(a⃗−b⃗)⋅(a⃗−b⃗)|\vec{a} - \vec{b}|^2 = (\vec{a} - \vec{b}) \cdot (\vec{a} - \vec{b})
step 8
Expand the dot product to get ∣a⃗−b⃗∣2=a⃗⋅a⃗−2a⃗⋅b⃗+b⃗⋅b⃗|\vec{a} - \vec{b}|^2 = \vec{a} \cdot \vec{a} - 2\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b}
step 9
Substitute a⃗⋅b⃗\vec{a} \cdot \vec{b} from step 5 into the equation from step 8 to find ∣a⃗−b⃗∣2|\vec{a} - \vec{b}|^2
step 10
Solve for ∣a⃗−b⃗∣|\vec{a} - \vec{b}| to find the range of values it can take
Answer
[Insert final answer here]
Key Concept
Vector properties and dot product
Explanation
The problem involves understanding vector properties, specifically the dot product, and how it relates to the magnitude of vectors and the angle between them. By manipulating the given equations, we can find the range of values for the magnitude of the difference between two vectors.
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