ABCD is a quadrilateral whose diagonals are AC and BD. Which one of the following is correct?
Analyzing the Quadrilateral Vector Problem
The question asks us to identify the correct vector equation among the given options for a quadrilateral ABCD with diagonals AC and BD. This involves understanding how vectors representing the sides and diagonals of a quadrilateral relate to each other through vector addition and subtraction principles.
In vector geometry, the vector from point X to point Y is denoted by \(\overrightarrow{XY}\). Vectors can be added using the triangle law or polygon law, and subtraction can be thought of as adding the negative of a vector (e.g., \(\overrightarrow{XY} - \overrightarrow{ZY} = \overrightarrow{XY} + \overrightarrow{YZ} = \overrightarrow{XZ}\) or simply \(\overrightarrow{XY} - \overrightarrow{XW} = \overrightarrow{WY}\)). Also, \(\overrightarrow{XY} = -\overrightarrow{YX}\).
Verifying the Correct Vector Equation for Quadrilateral ABCD
We need to test each option to see which vector equation holds true for any arbitrary quadrilateral ABCD. Let's examine the option that is stated as correct:
Option 2: \(\overrightarrow {{\rm{BA}}} + \overrightarrow {{\rm{CD}}} = \overrightarrow {{\rm{BD}}} + \overrightarrow {{\rm{CA}}} \)
To verify this equation, we can rearrange it to bring all terms to one side and check if the sum equals the zero vector \(\overrightarrow{0}\). Starting with the equation:
\(\overrightarrow {{\rm{BA}}} + \overrightarrow {{\rm{CD}}} = \overrightarrow {{\rm{BD}}} + \overrightarrow {{\rm{CA}}} \)
Subtract \(\overrightarrow {{\rm{BD}}}\) and \(\overrightarrow {{\rm{CA}}}\) from both sides:
\(\overrightarrow {{\rm{BA}}} + \overrightarrow {{\rm{CD}}} - \overrightarrow {{\rm{BD}}} - \overrightarrow {{\rm{CA}}} = \overrightarrow{0} \)
Using the property \(\overrightarrow{XY} = -\overrightarrow{YX}\), we can rewrite the subtracted terms:
- \(-\overrightarrow {{\rm{BD}}} = \overrightarrow {{\rm{DB}}}\)
- \(-\overrightarrow {{\rm{CA}}} = \overrightarrow {{\rm{AC}}}\)
Substitute these into the equation:
\(\overrightarrow {{\rm{BA}}} + \overrightarrow {{\rm{CD}}} + \overrightarrow {{\rm{DB}}} + \overrightarrow {{\rm{AC}}} = \overrightarrow{0} \)
Now, let's rearrange the terms on the left-hand side to apply the vector addition (polygon) law. We can group terms that form a path:
LHS = \(\overrightarrow {{\rm{BA}}} + \overrightarrow {{\rm{AC}}} + \overrightarrow {{\rm{CD}}} + \overrightarrow {{\rm{DB}}} \)
Applying the triangle law or polygon law step-by-step:
- \(\overrightarrow {{\rm{BA}}} + \overrightarrow {{\rm{AC}}}\) represents the sum of vectors going from B to A and then A to C. By the triangle law in triangle BAC, this sum is equal to \(\overrightarrow {{\rm{BC}}}\).
\(\overrightarrow {{\rm{BA}}} + \overrightarrow {{\rm{AC}}} = \overrightarrow {{\rm{BC}}}\) - Now substitute this back into the LHS: \(\overrightarrow {{\rm{BC}}} + \overrightarrow {{\rm{CD}}} + \overrightarrow {{\rm{DB}}} \)
- \(\overrightarrow {{\rm{BC}}} + \overrightarrow {{\rm{CD}}}\) represents the sum of vectors going from B to C and then C to D. By the triangle law in triangle BCD, this sum is equal to \(\overrightarrow {{\rm{BD}}}\).
\(\overrightarrow {{\rm{BC}}} + \overrightarrow {{\rm{CD}}} = \overrightarrow {{\rm{BD}}}\) - Now substitute this back into the expression: \(\overrightarrow {{\rm{BD}}} + \overrightarrow {{\rm{DB}}} \)
- \(\overrightarrow {{\rm{BD}}} + \overrightarrow {{\rm{DB}}}\) represents the sum of vectors going from B to D and then D back to B. This is the vector starting and ending at the same point B, which is the zero vector \(\overrightarrow{0}\).
\(\overrightarrow {{\rm{BD}}} + \overrightarrow {{\rm{DB}}} = \overrightarrow {{\rm{BB}}} = \overrightarrow{0}\)
So, the left-hand side simplifies to \(\overrightarrow{0}\).
Since the rearranged equation \(\overrightarrow {{\rm{BA}}} + \overrightarrow {{\rm{CD}}} + \overrightarrow {{\rm{DB}}} + \overrightarrow {{\rm{AC}}} \) simplifies to \(\overrightarrow{0}\), the original equation \(\overrightarrow {{\rm{BA}}} + \overrightarrow {{\rm{CD}}} = \overrightarrow {{\rm{BD}}} + \overrightarrow {{\rm{CA}}} \) is correct and holds true for any quadrilateral ABCD.
Why Other Options Are Incorrect
Let's briefly consider why the other options are generally incorrect for an arbitrary quadrilateral ABCD. If we rearranged any of the other options in the same manner (bringing all terms to one side), the resulting vector sum would not simplify to the zero vector \(\overrightarrow{0}\) for all quadrilaterals. For example, applying vector properties to Option 1 (\(\overrightarrow {{\rm{BA}}} + \overrightarrow {{\rm{CD}}} = \overrightarrow {{\rm{AC}}} + \overrightarrow {{\rm{DB}}} \)) would lead to an equation that is only true under specific conditions (like the quadrilateral being a parallelogram, though even that requires careful checking), not for a general quadrilateral.
Revision Table: Key Vector Properties Used
| Property | Description | Mathematical Form |
|---|---|---|
| Vector Addition (Triangle Law) | If three points A, B, and C form a triangle, the sum of two sides is the third side (when taken in order). | \(\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}\) |
| Vector Addition (Polygon Law) | For points \(A_1, A_2, \dots, A_n\), the sum of vectors forming a path is the vector connecting the start and end points. For a closed polygon, the sum is the zero vector. | \(\overrightarrow{A_1A_2} + \overrightarrow{A_2A_3} + \dots + \overrightarrow{A_{n-1}A_n} = \overrightarrow{A_1A_n}\) For a closed loop \(\overrightarrow{A_1A_2} + \dots + \overrightarrow{A_nA_1} = \overrightarrow{0}\) |
| Negative of a Vector | The vector from B to A is the negative of the vector from A to B. | \(\overrightarrow{BA} = -\overrightarrow{AB}\) |
| Zero Vector | A vector representing zero displacement. The sum of a vector and its negative. | \(\overrightarrow{AA} = \overrightarrow{0}\) \(\overrightarrow{AB} + \overrightarrow{BA} = \overrightarrow{0}\) |
Additional Information on Vector Geometry
Vector geometry is a powerful tool for solving geometric problems using vector algebra. Instead of relying on coordinates, we use the properties of vectors like magnitude and direction. The ability to add and subtract vectors allows us to express relationships between points and lines in space.
In any quadrilateral ABCD, the sides can be represented by vectors \(\overrightarrow{AB}\), \(\overrightarrow{BC}\), \(\overrightarrow{CD}\), and \(\overrightarrow{DA}\). The diagonals are represented by vectors \(\overrightarrow{AC}\) and \(\overrightarrow{BD}\).
Many geometric theorems can be proven elegantly using vectors. For example, the fact that the diagonals of a parallelogram bisect each other can be shown using vector addition.
The key takeaway from this problem is that specific vector relationships hold true for geometric figures like quadrilaterals, derived directly from the fundamental rules of vector addition and subtraction, regardless of the specific shape or position of the quadrilateral in space.
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