Consider the following statements in respect of a vector \(\vec c=\vec a+\vec b\) , where \(|\vec a|=|\vec b|\ne0\) : 1. \(\vec c\) is perpendicular to \((\vec a-\vec b).\) 2. \(\vec c\) is perpendicular to \(\vec a \times \vec b.\) Which of the above statement is/are correct?
Both 1 and 2
Analysing Vector Properties with Equal Magnitudes
The question asks us to examine the properties of a vector \(\vec c = \vec a + \vec b\), given that the magnitudes of vectors \(\vec a\) and \(\vec b\) are equal and non-zero, i.e., \(|\vec a| = |\vec b| \ne 0\). We need to verify two statements regarding the perpendicularity of \(\vec c\) to other vectors.
Statement 1: \(\vec c\) is perpendicular to \((\vec a - \vec b)\)
Two vectors are perpendicular if their dot product is zero. To check if \(\vec c\) is perpendicular to \((\vec a - \vec b)\), we compute the dot product \(\vec c \cdot (\vec a - \vec b)\).
Substitute \(\vec c = \vec a + \vec b\):
\( \vec c \cdot (\vec a - \vec b) = (\vec a + \vec b) \cdot (\vec a - \vec b) \)
Using the distributive property of the dot product:
\( (\vec a + \vec b) \cdot (\vec a - \vec b) = \vec a \cdot \vec a - \vec a \cdot \vec b + \vec b \cdot \vec a - \vec b \cdot \vec b \)
Recall that \(\vec a \cdot \vec a = |\vec a|^2\), \(\vec b \cdot \vec b = |\vec b|^2\), and the dot product is commutative (\(\vec a \cdot \vec b = \vec b \cdot \vec a\)):
\( \vec c \cdot (\vec a - \vec b) = |\vec a|^2 - \vec a \cdot \vec b + \vec a \cdot \vec b - |\vec b|^2 \)
Simplifying the expression:
\( \vec c \cdot (\vec a - \vec b) = |\vec a|^2 - |\vec b|^2 \)
Given that \(|\vec a| = |\vec b|\), we have \(|\vec a|^2 = |\vec b|^2\). Therefore,
\( \vec c \cdot (\vec a - \vec b) = |\vec a|^2 - |\vec a|^2 = 0 \)
Since the dot product is zero, \(\vec c\) is indeed perpendicular to \((\vec a - \vec b)\). Thus, Statement 1 is correct.
This property is related to the diagonals of a parallelogram. If \(|\vec a| = |\vec b|\), the parallelogram formed by \(\vec a\) and \(\vec b\) is a rhombus. The vector \(\vec c = \vec a + \vec b\) represents one diagonal, and \((\vec a - \vec b)\) represents the other diagonal. The diagonals of a rhombus are perpendicular.
Statement 2: \(\vec c\) is perpendicular to \((\vec a \times \vec b)\)
To check if \(\vec c\) is perpendicular to \((\vec a \times \vec b)\), we compute the dot product \(\vec c \cdot (\vec a \times \vec b)\).
Substitute \(\vec c = \vec a + \vec b\):
\( \vec c \cdot (\vec a \times \vec b) = (\vec a + \vec b) \cdot (\vec a \times \vec b) \)
Using the distributive property of the dot product:
\( (\vec a + \vec b) \cdot (\vec a \times \vec b) = \vec a \cdot (\vec a \times \vec b) + \vec b \cdot (\vec a \times \vec b) \)
Recall the property of the scalar triple product: \(\vec u \cdot (\vec v \times \vec w)\) represents the volume of the parallelepiped formed by \(\vec u, \vec v, \vec w\). If any two vectors are the same or parallel, the volume is zero.
- The term \(\vec a \cdot (\vec a \times \vec b)\) is a scalar triple product with the vector \(\vec a\) appearing twice. Thus, \(\vec a \cdot (\vec a \times \vec b) = 0\). This is also because the cross product \((\vec a \times \vec b)\) is always perpendicular to the vector \(\vec a\).
- Similarly, the term \(\vec b \cdot (\vec a \times \vec b)\) is a scalar triple product with the vector \(\vec b\) appearing twice (by rearranging the terms, it can be seen as \(\vec b \cdot (\vec a \times \vec b)\) which is equivalent to \((\vec b \times \vec a) \cdot \vec b\), or simply recognizing \((\vec a \times \vec b)\) is perpendicular to \(\vec b\)). Thus, \(\vec b \cdot (\vec a \times \vec b) = 0\).
Therefore,
\( \vec c \cdot (\vec a \times \vec b) = 0 + 0 = 0 \)
Since the dot product is zero, \(\vec c\) is perpendicular to \((\vec a \times \vec b)\). Thus, Statement 2 is correct.
The vector \((\vec a \times \vec b)\) is perpendicular to the plane containing \(\vec a\) and \(\vec b\). The vector \(\vec c = \vec a + \vec b\) lies in the plane formed by \(\vec a\) and \(\vec b\) (assuming they are not collinear). Any vector in a plane is perpendicular to a vector that is perpendicular to the plane.
Conclusion
Based on the analysis of both statements:
- Statement 1 is correct because \(\vec c \cdot (\vec a - \vec b) = |\vec a|^2 - |\vec b|^2 = 0\) when \(|\vec a| = |\vec b|\).
- Statement 2 is correct because \(\vec c \cdot (\vec a \times \vec b) = \vec a \cdot (\vec a \times \vec b) + \vec b \cdot (\vec a \times \vec b) = 0 + 0 = 0\).
Both statements are correct.
| Statement | Vector Operation | Result | Conclusion |
|---|---|---|---|
| 1 | \(\vec c \cdot (\vec a - \vec b)\) | \(|\vec a|^2 - |\vec b|^2\) | 0 since \(|\vec a| = |\vec b|\) \(\implies\) Perpendicular |
| 2 | \(\vec c \cdot (\vec a \times \vec b)\) | \(\vec a \cdot (\vec a \times \vec b) + \vec b \cdot (\vec a \times \vec b)\) | 0 + 0 = 0 \(\implies\) Perpendicular |
Revision Table: Vector Properties & Perpendicularity
| Concept | Description | Mathematical Representation |
|---|---|---|
| Dot Product | A scalar quantity representing the projection of one vector onto another, multiplied by the magnitude of the other vector. | \(\vec a \cdot \vec b = |\vec a| |\vec b| \cos \theta\) |
| Perpendicular Vectors | Two non-zero vectors are perpendicular if the angle between them is 90 degrees. Their dot product is zero. | \(\vec a \perp \vec b \iff \vec a \cdot \vec b = 0\) |
| Cross Product | A vector quantity perpendicular to the plane containing the two vectors. Its magnitude equals the area of the parallelogram formed by them. | \(|\vec a \times \vec b| = |\vec a| |\vec b| \sin \theta\) |
| Vector Perpendicular to Plane | The cross product \(\vec a \times \vec b\) is perpendicular to both \(\vec a\) and \(\vec b\), and any linear combination like \(x\vec a + y\vec b\) that lies in their plane (unless \(x=y=0\)). | \((\vec a \times \vec b) \cdot \vec a = 0\), \((\vec a \times \vec b) \cdot \vec b = 0\) |
| Scalar Triple Product | The dot product of one vector with the cross product of two others. Geometrically, it is the volume of the parallelepiped spanned by the three vectors. | \(\vec a \cdot (\vec b \times \vec c)\) |
Additional Information: Geometric Interpretation
When \(|\vec a| = |\vec b|\), the vectors \(\vec a\) and \(\vec b\) form two adjacent sides of a rhombus (or a square if they are perpendicular). The vector \(\vec c = \vec a + \vec b\) represents the main diagonal of this rhombus/square, starting from the common origin of \(\vec a\) and \(\vec b\). The vector \((\vec a - \vec b)\) represents the other diagonal, pointing from the tip of \(\vec b\) to the tip of \(\vec a\).
- Statement 1 Geometric Meaning: The diagonals of a rhombus are perpendicular. This matches our mathematical result \(\vec c \cdot (\vec a - \vec b) = 0\).
- Statement 2 Geometric Meaning: The vector \((\vec a \times \vec b)\) is perpendicular to the plane containing \(\vec a\) and \(\vec b\). Since \(\vec c = \vec a + \vec b\) lies in the same plane as \(\vec a\) and \(\vec b\), it must be perpendicular to any vector that is perpendicular to that plane, such as \((\vec a \times \vec b)\). This matches our mathematical result \(\vec c \cdot (\vec a \times \vec b) = 0\).
Both interpretations confirm the mathematical findings that both statements are correct under the given condition \(|\vec a| = |\vec b| \ne 0\).
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Select the correct answer using the code given below.
Let \(\vec{a}\), \(\vec{b}\), \(\vec{c}\) and \(\vec{d}\) be the vectors. Consider the following:
I. \((\vec{a}\times\vec{b})\cdot(\vec{c}\times\vec{d})\)
II. \((\vec{a}\times\vec{b})\times(\vec{c}\times\vec{d})\)
III. \((\vec{a}\cdot\vec{b})\cdot(\vec{c}\cdot\vec{d})\)
IV. \((\vec{a}\cdot\vec{b})\times(\vec{c}\cdot\vec{d})\)
V. \(\{(\vec{a}\times\vec{b})\cdot\vec{c}\}\times\vec{d}\)
where '\(\cdot\)' represents scalar product of vectors and '\(\times\)' represents vector product of vectors. How many of the above are not well-defined?
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If the volume of a parallelepiped whose adjacent edges are
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