Let \(\rm \vec{a}, \vec{b}\) and \(\rm\vec{c}\) be unit vectors such that \(\rm\vec{a} \times \vec{b}\) is perpendicular to \(\vec{c}\) . If θ is the angle between \(\rm\vec{a}\) and \(\rm\vec{b}\) , then which of the following is/are correct? 1. \(\rm\vec{a} \times \vec{b} = sin ~\theta~ \vec{c} \) 2. \(\rm\vec {a} \cdot (\vec{b}\times \vec{c})=0\) Select the correct answer using the code given below.
2 only
Analyzing Vector Properties with Unit Vectors
The question provides information about three unit vectors, \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\), and a condition regarding their cross and dot products. We need to determine which of the given statements is/are correct based on these conditions.
Let's list the given information:
- \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are unit vectors. This means \(|\vec{a}| = 1\), \(|\vec{b}| = 1\), and \(|\vec{c}| = 1\).
- \(\vec{a} \times \vec{b}\) is perpendicular to \(\vec{c}\). The condition of perpendicularity between two vectors means their dot product is zero. So, \((\vec{a} \times \vec{b}) \cdot \vec{c} = 0\).
- \(\theta\) is the angle between \(\vec{a}\) and \(\vec{b}\), where \(0 \le \theta \le \pi\).
Now let's analyze each statement.
Statement 1: \(\vec{a} \times \vec{b} = \sin ~\theta~ \vec{c}\)
Let's evaluate this statement by considering both magnitude and direction.
Magnitude Check:
- The magnitude of the cross product \(\vec{a} \times \vec{b}\) is given by \(|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta\). Since \(\vec{a}\) and \(\vec{b}\) are unit vectors, \(|\vec{a} \times \vec{b}| = (1)(1)\sin \theta = \sin \theta\). (Note that since \(0 \le \theta \le \pi\), \(\sin \theta \ge 0\), so \(|\sin \theta| = \sin \theta\)).
- The magnitude of the vector \(\sin \theta \vec{c}\) is \(|\sin \theta \vec{c}| = |\sin \theta| |\vec{c}|\). Since \(\vec{c}\) is a unit vector and \(\sin \theta \ge 0\), \(|\sin \theta \vec{c}| = \sin \theta \cdot 1 = \sin \theta\).
The magnitudes match: \(|\vec{a} \times \vec{b}| = |\sin \theta \vec{c}| = \sin \theta\). This part of the statement is consistent with the definition of the cross product magnitude.
Direction Check:
- The direction of the vector \(\vec{a} \times \vec{b}\) is perpendicular to the plane containing \(\vec{a}\) and \(\vec{b}\).
- The direction of the vector \(\sin \theta \vec{c}\) is parallel to \(\vec{c}\) (assuming \(\sin \theta \ne 0\)). If \(\sin \theta = 0\), then both sides are the zero vector, and the equality holds.
For the vector equality \(\vec{a} \times \vec{b} = \sin \theta \vec{c}\) to hold (when \(\sin \theta \ne 0\)), the direction of \(\vec{a} \times \vec{b}\) must be parallel to the direction of \(\vec{c}\). However, we are given that \(\vec{a} \times \vec{b}\) is perpendicular to \(\vec{c}\), which means their dot product is zero: \((\vec{a} \times \vec{b}) \cdot \vec{c} = 0\).
If statement 1 were true, then we would have:
\((\vec{a} \times \vec{b}) \cdot \vec{c} = (\sin \theta \vec{c}) \cdot \vec{c}\)
\(0 = \sin \theta (\vec{c} \cdot \vec{c})\)
\(0 = \sin \theta |\vec{c}|^2\)
\(0 = \sin \theta (1)^2\)
\(0 = \sin \theta\)
This implies that if statement 1 is true, then \(\sin \theta\) must be 0. This happens only when \(\theta = 0\) or \(\theta = \pi\). The question states that \(\theta\) is the angle between \(\vec{a}\) and \(\vec{b}\), which can be any value in \([0, \pi]\), not necessarily 0 or \(\pi\). The given condition \((\vec{a} \times \vec{b}) \cdot \vec{c} = 0\) means \(\vec{a} \times \vec{b}\) is perpendicular to \(\vec{c}\). Statement 1 means \(\vec{a} \times \vec{b}\) is parallel to \(\vec{c}\) (when \(\sin\theta \ne 0\)). A non-zero vector cannot be simultaneously parallel and perpendicular to another non-zero vector. Thus, statement 1 is only true in the special case where \(\vec{a} \times \vec{b} = \vec{0}\) (i.e., \(\sin \theta = 0\)), which is not guaranteed by the problem statement. Therefore, statement 1 is not correct in general.
Statement 2: \(\vec{a} \cdot (\vec{b}\times \vec{c})=0\)
This expression \(\vec{a} \cdot (\vec{b} \times \vec{c})\) is the scalar triple product of the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\). A key property of the scalar triple product is its cyclic permutation property:
\(\vec{a} \cdot (\vec{b} \times \vec{c}) = \vec{b} \cdot (\vec{c} \times \vec{a}) = \vec{c} \cdot (\vec{a} \times \vec{b})\)
Also, the dot and cross can be swapped, though this is usually written by rearranging terms first, e.g., \(\vec{c} \cdot (\vec{a} \times \vec{b}) = (\vec{a} \times \vec{b}) \cdot \vec{c}\).
So, the expression in statement 2, \(\vec{a} \cdot (\vec{b} \times \vec{c})\), is equal to \((\vec{a} \times \vec{b}) \cdot \vec{c}\).
We are given that \(\vec{a} \times \vec{b}\) is perpendicular to \(\vec{c}\), which means \((\vec{a} \times \vec{b}) \cdot \vec{c} = 0\).
Since \(\vec{a} \cdot (\vec{b} \times \vec{c}) = (\vec{a} \times \vec{b}) \cdot \vec{c}\), and \((\vec{a} \times \vec{b}) \cdot \vec{c} = 0\), it directly follows that \(\vec{a} \cdot (\vec{b} \times \vec{c}) = 0\).
Therefore, statement 2 is correct based on the given conditions.
Summary of Analysis
- Statement 1: \(\vec{a} \times \vec{b} = \sin ~\theta~ \vec{c}\) is not generally true under the given conditions. It would require \(\vec{a} \times \vec{b}\) to be parallel to \(\vec{c}\) (or both zero), which contradicts the condition that they are perpendicular (unless both are zero).
- Statement 2: \(\vec{a} \cdot (\vec{b}\times \vec{c})=0\) is true because it is equivalent to the given condition \((\vec{a} \times \vec{b}) \cdot \vec{c} = 0\) due to the properties of the scalar triple product.
Based on this analysis, only statement 2 is correct.
| Statement | Analysis | Correctness |
|---|---|---|
| 1. \(\vec{a} \times \vec{b} = \sin ~\theta~ \vec{c}\) | Requires \(\vec{a} \times \vec{b}\) to be parallel to \(\vec{c}\) (if \(\sin \theta \ne 0\)), which contradicts the given condition of perpendicularity unless \(\vec{a} \times \vec{b} = \vec{0}\). | Incorrect (in general) |
| 2. \(\vec{a} \cdot (\vec{b}\times \vec{c})=0\) | Equivalent to \((\vec{a} \times \vec{b}) \cdot \vec{c} = 0\) by scalar triple product properties, which is the given condition. | Correct |
Therefore, only statement 2 is correct.
Revision Table: Key Vector Concepts
| Concept | Description | Mathematical Representation |
|---|---|---|
| Unit Vector | A vector with a magnitude of 1. | \(|\vec{v}| = 1\) |
| Cross Product (\(\vec{a} \times \vec{b}\)) | A vector perpendicular to both \(\vec{a}\) and \(\vec{b}\). Its magnitude is \(|\vec{a}||\vec{b}|\sin\theta\), where \(\theta\) is the angle between \(\vec{a}\) and \(\vec{b}\). | \(\vec{a} \times \vec{b}\) is perpendicular to \(\vec{a}\) and \(\vec{b}\). \(|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta\) |
| Dot Product (\(\vec{a} \cdot \vec{b}\)) | A scalar quantity. If \(\vec{a}\) and \(\vec{b}\) are perpendicular, their dot product is 0. | \(\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\phi\). If \(\vec{a} \perp \vec{b}\), then \(\vec{a} \cdot \vec{b} = 0\). |
| Scalar Triple Product (\(\vec{a} \cdot (\vec{b} \times \vec{c})\)) | A scalar quantity representing the volume of the parallelepiped formed by \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\). It is zero if the vectors are coplanar. | \(\vec{a} \cdot (\vec{b} \times \vec{c}) = [\vec{a}, \vec{b}, \vec{c}]\). Property: \(\vec{a} \cdot (\vec{b} \times \vec{c}) = (\vec{a} \times \vec{b}) \cdot \vec{c}\). |
| Perpendicular Vectors | Two non-zero vectors are perpendicular if the angle between them is \(90^\circ\) (\(\pi/2\) radians). Their dot product is zero. | \(\vec{u} \perp \vec{v} \implies \vec{u} \cdot \vec{v} = 0\) (for \(\vec{u} \ne \vec{0}, \vec{v} \ne \vec{0}\)) |
Additional Information: Coplanarity and Scalar Triple Product
The scalar triple product \(\vec{a} \cdot (\vec{b} \times \vec{c})\) has a geometric interpretation. It represents the signed volume of the parallelepiped formed by the three vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) when they are placed with a common initial point.
If the scalar triple product is zero, it means the volume of the parallelepiped is zero. This can only happen if the three vectors lie in the same plane. Therefore, \(\vec{a} \cdot (\vec{b} \times \vec{c}) = 0\) is the condition for the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) to be coplanar.
The given condition states that \(\vec{a} \times \vec{b}\) is perpendicular to \(\vec{c}\). The vector \(\vec{a} \times \vec{b}\) is itself perpendicular to the plane containing \(\vec{a}\) and \(\vec{b}\) (assuming \(\vec{a}\) and \(\vec{b}\) are not parallel). If this vector \(\vec{a} \times \vec{b}\) is perpendicular to \(\vec{c}\), it means \(\vec{c}\) must lie in the plane spanned by \(\vec{a}\) and \(\vec{b}\). This confirms that \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are coplanar, which is equivalent to their scalar triple product being zero: \(\vec{a} \cdot (\vec{b} \times \vec{c}) = 0\).
This geometric interpretation provides further support for why statement 2 must be correct given the condition that \(\vec{a} \times \vec{b}\) is perpendicular to \(\vec{c}\).
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