Consider the following statements: 1. The cross product of two unit vectors is always a unit vector. 2. The dot product of two unit vectors is always unity. 3. The magnitude of sum of two unit vectors is always greater than the magnitude of their difference. Which of the above statements are not correct?
1, 2 and 3
Analyzing Statements on Unit Vector Properties
The question asks us to evaluate three statements regarding unit vectors and identify which ones are not correct. A unit vector is a vector with a magnitude of 1.
Statement 1: The cross product of two unit vectors is always a unit vector.
Let \(\mathbf{\hat{u}}\) and \(\mathbf{\hat{v}}\) be two unit vectors. The magnitude of their cross product is given by:
\( |\mathbf{\hat{u}} \times \mathbf{\hat{v}}| = |\mathbf{\hat{u}}| |\mathbf{\hat{v}}| \sin(\theta) \)
where \(\theta\) is the angle between the two vectors. Since \(\mathbf{\hat{u}}\) and \(\mathbf{\hat{v}}\) are unit vectors, \(|\mathbf{\hat{u}}| = 1\) and \(|\mathbf{\hat{v}}| = 1\).
So, the magnitude becomes:
\( |\mathbf{\hat{u}} \times \mathbf{\hat{v}}| = (1)(1)\sin(\theta) = \sin(\theta) \)
For the cross product \(\mathbf{\hat{u}} \times \mathbf{\hat{v}}\) to be a unit vector, its magnitude must be 1. This requires \(\sin(\theta) = 1\). The sine function is 1 only when \(\theta = 90^\circ\) (or \(\pi/2\) radians).
If the angle \(\theta\) is anything other than \(90^\circ\) (e.g., \(30^\circ\), \(45^\circ\), or \(0^\circ\), \(180^\circ\)), the magnitude \(\sin(\theta)\) will be less than 1 (or 0 if \(\theta = 0^\circ\) or \(180^\circ\)). A vector with magnitude 0 is the zero vector, which is not a unit vector. A vector with magnitude less than 1 is also not a unit vector.
Therefore, the cross product of two unit vectors is not always a unit vector.
Statement 1 is not correct.
Statement 2: The dot product of two unit vectors is always unity.
Let \(\mathbf{\hat{u}}\) and \(\mathbf{\hat{v}}\) be two unit vectors. The dot product is given by:
\( \mathbf{\hat{u}} \cdot \mathbf{\hat{v}} = |\mathbf{\hat{u}}| |\mathbf{\hat{v}}| \cos(\theta) \)
where \(\theta\) is the angle between the two vectors. Since \(\mathbf{\hat{u}}\) and \(\mathbf{\hat{v}}\) are unit vectors, \(|\mathbf{\hat{u}}| = 1\) and \(|\mathbf{\hat{v}}| = 1\).
So, the dot product becomes:
\( \mathbf{\hat{u}} \cdot \mathbf{\hat{v}} = (1)(1)\cos(\theta) = \cos(\theta) \)
For the dot product to be unity (1), \(\cos(\theta) = 1\). The cosine function is 1 only when \(\theta = 0^\circ\). This occurs when the two unit vectors are parallel and point in the same direction.
If the angle \(\theta\) is different (e.g., \(90^\circ\)), \(\cos(90^\circ) = 0\). The dot product is 0. If \(\theta = 180^\circ\) (antiparallel), \(\cos(180^\circ) = -1\). The dot product is -1.
The dot product of two unit vectors can take any value between -1 and 1, depending on the angle between them. It is not always unity.
Therefore, the dot product of two unit vectors is not always unity.
Statement 2 is not correct.
Statement 3: The magnitude of sum of two unit vectors is always greater than the magnitude of their difference.
Let \(\mathbf{\hat{u}}\) and \(\mathbf{\hat{v}}\) be two unit vectors with an angle \(\theta\) between them. The magnitudes of their sum and difference are given by:
Magnitude of sum: \(|\mathbf{\hat{u}} + \mathbf{\hat{v}}| = \sqrt{|\mathbf{\hat{u}}|^2 + |\mathbf{\hat{v}}|^2 + 2|\mathbf{\hat{u}}||\mathbf{\hat{v}}|\cos(\theta)}\)
Since \(|\mathbf{\hat{u}}|=1\) and \(|\mathbf{\hat{v}}|=1\), this becomes:
\( |\mathbf{\hat{u}} + \mathbf{\hat{v}}| = \sqrt{1^2 + 1^2 + 2(1)(1)\cos(\theta)} = \sqrt{2 + 2\cos(\theta)} \)
Magnitude of difference: \(|\mathbf{\hat{u}} - \mathbf{\hat{v}}| = \sqrt{|\mathbf{\hat{u}}|^2 + |\mathbf{\hat{v}}|^2 - 2|\mathbf{\hat{u}}||\mathbf{\hat{v}}|\cos(\theta)}\)
Since \(|\mathbf{\hat{u}}|=1\) and \(|\mathbf{\hat{v}}|=1\), this becomes:
\( |\mathbf{\hat{u}} - \mathbf{\hat{v}}| = \sqrt{1^2 + 1^2 - 2(1)(1)\cos(\theta)} = \sqrt{2 - 2\cos(\theta)} \)
Let's consider some examples based on the angle \(\theta\):
- Case 1: Vectors are parallel (\(\theta = 0^\circ\))
- \(|\mathbf{\hat{u}} + \mathbf{\hat{v}}| = \sqrt{2 + 2\cos(0^\circ)} = \sqrt{2 + 2(1)} = \sqrt{4} = 2\)
- \(|\mathbf{\hat{u}} - \mathbf{\hat{v}}| = \sqrt{2 - 2\cos(0^\circ)} = \sqrt{2 - 2(1)} = \sqrt{0} = 0\)
- Here, \(2 > 0\). The sum magnitude is greater.
- Case 2: Vectors are perpendicular (\(\theta = 90^\circ\))
- \(|\mathbf{\hat{u}} + \mathbf{\hat{v}}| = \sqrt{2 + 2\cos(90^\circ)} = \sqrt{2 + 2(0)} = \sqrt{2}\)
- \(|\mathbf{\hat{u}} - \mathbf{\hat{v}}| = \sqrt{2 - 2\cos(90^\circ)} = \sqrt{2 - 2(0)} = \sqrt{2}\)
- Here, \(\sqrt{2}\) is not greater than \(\sqrt{2}\). The magnitudes are equal.
- Case 3: Vectors are antiparallel (\(\theta = 180^\circ\))
- \(|\mathbf{\hat{u}} + \mathbf{\hat{v}}| = \sqrt{2 + 2\cos(180^\circ)} = \sqrt{2 + 2(-1)} = \sqrt{0} = 0\)
- \(|\mathbf{\hat{u}} - \mathbf{\hat{v}}| = \sqrt{2 - 2\cos(180^\circ)} = \sqrt{2 - 2(-1)} = \sqrt{4} = 2\)
- Here, \(0\) is not greater than \(2\). The difference magnitude is greater.
The magnitude of the sum is not always greater than the magnitude of the difference. They can be equal or the difference magnitude can be greater.
Therefore, the magnitude of sum of two unit vectors is not always greater than the magnitude of their difference.
Statement 3 is not correct.
Summary of Statements
Based on our analysis:
- Statement 1 is incorrect.
- Statement 2 is incorrect.
- Statement 3 is incorrect.
All three statements are not correct.
| Statement | Correct? | Reason |
|---|---|---|
| Cross product is always unit vector | No | Magnitude depends on \(\sin(\theta)\); equals 1 only if \(\theta=90^\circ\). |
| Dot product is always unity | No | Result depends on \(\cos(\theta)\); equals 1 only if \(\theta=0^\circ\). Can be between -1 and 1. |
| Magnitude of sum > Magnitude of difference | No | Depends on angle \(\theta\). Can be equal (\(\theta=90^\circ\)) or less (\(\theta > 90^\circ\)). |
The statements that are not correct are 1, 2, and 3.
Revision Table: Key Unit Vector Properties
| Operation | Formula (for unit vectors \(\mathbf{\hat{u}}, \mathbf{\hat{v}}\) with angle \(\theta\)) | Possible Results/Magnitudes |
|---|---|---|
| Cross Product Magnitude (\(|\mathbf{\hat{u}} \times \mathbf{\hat{v}}|\)) | \(|\mathbf{\hat{u}}| |\mathbf{\hat{v}}| \sin(\theta) = \sin(\theta)\) | Between 0 and 1. Equals 1 only if \(\theta=90^\circ\). |
| Dot Product (\(\mathbf{\hat{u}} \cdot \mathbf{\hat{v}}\)) | \(|\mathbf{\hat{u}}| |\mathbf{\hat{v}}| \cos(\theta) = \cos(\theta)\) | Between -1 and 1. Equals 1 only if \(\theta=0^\circ\). |
| Sum Magnitude (\(|\mathbf{\hat{u}} + \mathbf{\hat{v}}|\)) | \(\sqrt{2 + 2\cos(\theta)}\) or \(2|\cos(\theta/2)|\) | Between 0 (for \(\theta=180^\circ\)) and 2 (for \(\theta=0^\circ\)). |
| Difference Magnitude (\(|\mathbf{\hat{u}} - \mathbf{\hat{v}}|\)) | \(\sqrt{2 - 2\cos(\theta)}\) or \(2|\sin(\theta/2)|\) | Between 0 (for \(\theta=0^\circ\)) and 2 (for \(\theta=180^\circ\)). |
Additional Information on Vector Operations
Understanding vector operations like the dot product and cross product is fundamental in physics and engineering. They describe different aspects of how vectors interact.
- Dot Product: The dot product results in a scalar quantity. It is related to the projection of one vector onto another and is used to calculate work done by a force, electrical potential, etc. If the dot product is zero, the vectors are orthogonal (perpendicular), assuming they are not zero vectors.
- Cross Product: The cross product results in a vector quantity that is perpendicular to both original vectors. Its magnitude is related to the area of the parallelogram formed by the vectors. It is used to calculate torque, magnetic force, etc. If the cross product is the zero vector, the original vectors are parallel or antiparallel.
- Vector Addition/Subtraction: These operations follow the parallelogram or triangle rules. The magnitude of the resulting vector depends heavily on the angle between the original vectors, as seen in the formulas for \(|\mathbf{\hat{u}} + \mathbf{\hat{v}}|\) and \(|\mathbf{\hat{u}} - \mathbf{\hat{v}}|\).
It is important to remember that the specific values of dot and cross products, and the magnitudes of sums/differences, depend on the relative orientation (the angle \(\theta\)) of the vectors involved, even if they are unit vectors.
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