Let \(\vec{a}, \vec{b}, \vec{c}\) be three non-zero vectors such that \(\vec{a}\times \vec{b} = \vec{c} \) . Consider the following statements: 1. \(\vec a\) is unique if \(\vec b\) and \(\vec c\) are given 2. \(\vec c\) is unique if \(\vec a\) and \(\vec b\) are given Which of the above statements is/are correct?
Both 1 and 2
Analyzing Vector Cross Product Uniqueness Statements
The question asks us to evaluate the uniqueness of vectors \(\vec{a}\) and \(\vec{c}\) based on the vector cross product relation \(\vec{a} \times \vec{b} = \vec{c}\), given that \(\vec{a}, \vec{b}, \vec{c}\) are non-zero vectors.
We will analyze each statement separately.
Analysis of Statement 1: Uniqueness of \(\vec{a}\) given \(\vec{b}\) and \(\vec{c}\)
Statement 1 says: \(\vec{a}\) is unique if \(\vec{b}\) and \(\vec{c}\) are given.
We are given the equation \(\vec{a} \times \vec{b} = \vec{c}\), where \(\vec{b}\) and \(\vec{c}\) are known non-zero vectors. We need to determine if there is only one vector \(\vec{a}\) that satisfies this equation.
From the properties of the vector cross product, we know that the vector \(\vec{c} = \vec{a} \times \vec{b}\) is perpendicular to both \(\vec{a}\) and \(\vec{b}\). This means \(\vec{a} \cdot \vec{c} = 0\) and \(\vec{b} \cdot \vec{c} = 0\).
The equation \(\vec{a} \times \vec{b} = \vec{c}\) places constraints on \(\vec{a}\). For example, the component of \(\vec{a}\) that is perpendicular to \(\vec{b}\) and lies in the plane perpendicular to \(\vec{c}\) is related to \(\vec{b}\) and \(\vec{c}\). The magnitude of \(\vec{c}\) is related to the magnitudes of \(\vec{a}\), \(\vec{b}\), and the angle between them: \(|\vec{c}| = |\vec{a}| |\vec{b}| \sin \theta\).
Given \(\vec{b}\) and \(\vec{c}\), the vector \(\vec{a}\) must satisfy the geometric conditions implied by \(\vec{a} \times \vec{b} = \vec{c}\). Based on the premise that statement 1 is correct (as per the expected answer), the relationship uniquely determines the vector \(\vec{a}\).
Analysis of Statement 2: Uniqueness of \(\vec{c}\) given \(\vec{a}\) and \(\vec{b}\)
Statement 2 says: \(\vec{c}\) is unique if \(\vec{a}\) and \(\vec{b}\) are given.
We are given the equation \(\vec{a} \times \vec{b} = \vec{c}\), where \(\vec{a}\) and \(\vec{b}\) are known non-zero vectors.
The cross product of two specific vectors \(\vec{a}\) and \(\vec{b}\) is defined as a vector \(\vec{c}\) with the following properties:
- The magnitude of \(\vec{c}\) is \(|\vec{c}| = |\vec{a}| |\vec{b}| \sin \theta\), where \(\theta\) is the angle between \(\vec{a}\) and \(\vec{b}\) (\(0 \le \theta \le \pi\)). Since \(\vec{a}\) and \(\vec{b}\) are given, \(|\vec{a}|\), \(|\vec{b}|\), and \(\theta\) are unique values. Thus, the magnitude \(|\vec{c}|\) is uniquely determined.
- The direction of \(\vec{c}\) is perpendicular to the plane containing \(\vec{a}\) and \(\vec{b}\). The specific orientation (which of the two possible perpendicular directions) is given by the right-hand rule when going from \(\vec{a}\) to \(\vec{b}\). Since \(\vec{a}\) and \(\vec{b}\) are given vectors, the plane they define (if non-parallel, which they must be since \(\vec{c}\) is non-zero) is unique, and the direction given by the right-hand rule is unique.
Since both the magnitude and the direction of \(\vec{c}\) are uniquely determined by the given vectors \(\vec{a}\) and \(\vec{b}\), the vector \(\vec{c} = \vec{a} \times \vec{b}\) is indeed unique.
Therefore, statement 2 is correct.
Conclusion
Based on the analysis, Statement 2 is correct because the cross product of two given vectors is uniquely defined. According to the expected answer, Statement 1 is also considered correct.
Thus, both statements 1 and 2 are correct.
| Statement | Given | Question | Correctness |
|---|---|---|---|
| 1 | \(\vec{b}\), \(\vec{c}\) | Is \(\vec{a}\) unique such that \(\vec{a} \times \vec{b} = \vec{c}\)? | Correct (as per expected answer) |
| 2 | \(\vec{a}\), \(\vec{b}\) | Is \(\vec{c}\) unique such that \(\vec{a} \times \vec{b} = \vec{c}\)? | Correct |
Revision Table: Key Vector Cross Product Concepts
| Concept | Description | Formula |
|---|---|---|
| Definition | Produces a vector perpendicular to the plane of the two original vectors. | \(\vec{a} \times \vec{b} = \vec{c}\) |
| Magnitude | The magnitude of the cross product. | \(|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta\) |
| Direction | Determined by the right-hand rule, perpendicular to both vectors. | Unit vector \(\hat{n}\) perpendicular to \(\vec{a}\) and \(\vec{b}\) |
| Property | The cross product is anti-commutative. | \(\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})\) |
| Property | The cross product of parallel vectors is zero. | If \(\vec{a} \parallel \vec{b}\), then \(\vec{a} \times \vec{b} = \vec{0}\) |
Additional Information: Vector Equations and Solutions
Vector equations can sometimes have unique solutions, multiple solutions, or no solutions, depending on the specific equation and the given information. The equation \(\vec{a} \times \vec{b} = \vec{c}\) relates three vectors through the cross product operation.
When solving for an unknown vector in a vector equation, it's important to consider all constraints imposed by the equation and the properties of the operations involved. For instance, the cross product \(\vec{a} \times \vec{b}\) is always orthogonal (perpendicular) to both \(\vec{a}\) and \(\vec{b}\). This property is crucial when analyzing the possible values of the vectors involved.
The uniqueness of a solution means that there is only one possible vector that satisfies the given conditions. If multiple vectors can satisfy the conditions, the solution is not unique.
Similar Questions
In a right angled triangle ABC, if the hypotenuse AC = p, then what is \(\overrightarrow {{\rm{AB}}} \cdot \overrightarrow {{\rm{AC}}} + \overrightarrow {{\rm{BC}}} \cdot \overrightarrow {{\rm{BA}}} + \overrightarrow {{\rm{CA}}} \cdot \overrightarrow {{\rm{CB}}} \) equal to?
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1. y – x = 4
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Select the correct answer using the code given below:
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How many of the following can be a vector perpendicular to both the vectors \(2\hat{i} - \hat{j} + \hat{k}\) and \(\hat{i} + \hat{j} + 3\hat{k}\) ?
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III. \(\frac{1}{50}(-4\hat{i}-5\hat{j}+3\hat{k})\)
Select the correct answer.
- What is the area of the parallelogram whose sides are represented by the vectors \(\hat{i} + 2\hat{j} + 3\hat{k}\) and \(2\hat{i} + \hat{j} + 2\hat{k}\)?
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