Problem 12
Question
In a parallelogram \(A B C D,|A B|=a,|A D|=b\) and \(\mid A C\) \(\mid=c\). Then, \(D B \cdot A B\) has the value (A) \(\frac{3 a^{2}+b^{2}-c^{2}}{2}\) (B) \(\frac{a^{2}+3 b^{2}-c^{2}}{2}\) (C) \(\frac{a^{2}-b^{2}+3 c^{2}}{2}\) (D) \(\frac{a^{2}+3 b^{2}+c^{2}}{2}\)
Step-by-Step Solution
Verified Answer
The answer is (A) \(\frac{3a^2 + b^2 - c^2}{2}\).
1Step 1: Understand the Parallelogram Diagonal and Side Relationships
In the parallelogram \(ABCD\), we have sides \(|AB|=a\) and \(|AD|=b\), along with the diagonal \(|AC|=c\). We aim to find the value of the dot product \(DB \cdot AB\). In a parallelogram, the relation \(AC^2 + BD^2 = 2(AB^2 + AD^2)\) will be useful to determine \(DB\).
2Step 2: Express \(DB\) using the Diagonal and Side Property
Given that \(DB\) is the other diagonal, we apply the relation for diagonals, \(AC^2 + BD^2 = 2(AB^2 + AD^2)\), which gives:\[c^2 + BD^2 = 2(a^2 + b^2)\] Solving for \(BD^2\) gives us \(BD^2 = 2(a^2 + b^2) - c^2\).
3Step 3: Compute the Dot Product \(DB \cdot AB\)
To find \(DB \cdot AB\), recognize that the dot product formula involves both the magnitudes and the cosine of the angle between them. However, in terms of vectors, we use : \[DB \cdot AB = \frac{1}{2} (DB^2 + AB^2 - AD^2 )\]First, solve for \(DB^2\) from Step 2 to substitute into the formula: \[DB^2 = 2(a^2 + b^2) - c^2\]Substitute in the dot product formula: \[DB \cdot AB = \frac{1}{2} \left((2(a^2 + b^2) - c^2) + a^2 - b^2 \right)\]Simplify this to: \[DB \cdot AB = \frac{1}{2} (3a^2 + b^2 - c^2)\]
4Step 4: Choose the Correct Answer Option
The expression matches one of the provided answers: \(\frac{3a^2 + b^2 - c^2}{2}\). This corresponds to option (A).
Key Concepts
Parallelogram PropertiesDot ProductDiagonal Relationships
Parallelogram Properties
Parallelograms are fundamental shapes in vector algebra, characterized by opposite sides that are both equal and parallel. In a parallelogram, such as the one we are examining, several properties can guide us through our calculations.
Some essential properties include:
Understanding these properties not only aids in solving problems related to parallelograms but also enhances spatial reasoning skills within vector algebra.
Some essential properties include:
- Opposite sides being equal in length: \( |AB| = |CD| \) and \( |AD| = |BC| \).
- Opposite angles being equal.
- The diagonals bisecting each other.
- A special relationship between the lengths of the diagonals and the sides: \( AC^2 + BD^2 = 2(AB^2 + AD^2) \). This crucial property helps in determining unknown diagonal lengths when sides are known.
Understanding these properties not only aids in solving problems related to parallelograms but also enhances spatial reasoning skills within vector algebra.
Dot Product
The dot product, also known as the scalar product, is a fundamental operation in vector algebra that combines two vectors to yield a scalar. It provides a measure of the extent to which two vectors point in the same direction.
The dot product \( \mathbf{a} \cdot \mathbf{b} \) of vectors \( \mathbf{a} \) and \( \mathbf{b} \) is defined as:
\[\mathbf{a} \cdot \mathbf{b} = | \mathbf{a} | \, \times \, | \mathbf{b} | \cdot \cos \theta\]
Among its properties, the dot product is commutative and linear. It is especially useful when calculating angles between vectors or determining projections.
The dot product \( \mathbf{a} \cdot \mathbf{b} \) of vectors \( \mathbf{a} \) and \( \mathbf{b} \) is defined as:
\[\mathbf{a} \cdot \mathbf{b} = | \mathbf{a} | \, \times \, | \mathbf{b} | \cdot \cos \theta\]
- Here, \( | \mathbf{a} | \) and \( | \mathbf{b} | \) are the magnitudes of vectors \( \mathbf{a} \) and \( \mathbf{b} \) respectively.
- \( \theta \) is the angle between them.
- Alternatively, if the vectors are expressed in component form, the dot product can be calculated using their components.
Among its properties, the dot product is commutative and linear. It is especially useful when calculating angles between vectors or determining projections.
Diagonal Relationships
In a parallelogram, the relationship between the diagonals and the sides plays a crucial role in problem-solving. These relationships stem from the intrinsic geometric properties of a parallelogram.
One key relationship is the equation:
\[ AC^2 + BD^2 = 2(AB^2 + AD^2) \]
Understanding how the lengths of the diagonals interrelate with the sides of a parallelogram is vital for effectively solving related geometric problems.
One key relationship is the equation:
\[ AC^2 + BD^2 = 2(AB^2 + AD^2) \]
- This equation relates the squares of the lengths of diagonals \( AC \) and \( BD \) to the squares of the sides \( AB \) and \( AD \).
- The equation derives from the law of cosines applied within the various triangles formed by the diagonals.
- This form allows for straightforward algebraic manipulation to solve for unknowns, such as finding the length of one diagonal given the other diagonal and two side lengths.
Understanding how the lengths of the diagonals interrelate with the sides of a parallelogram is vital for effectively solving related geometric problems.
Other exercises in this chapter
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