Problem 12 - A Complete Resource Book in Mathematics for JEE Main 2017

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

Answer

The answer is (A) \( \frac{3a^2 + b^2 - c^2}{2} \).

Step 1: Understand the problem

We need to find the dot product of vector \( \overrightarrow{DB} \) with vector \( \overrightarrow{AB} \) in the parallelogram \( ABCD \). We have the side lengths: \( |AB| = a \), \( |AD| = b \), and the diagonal \( |AC| = c \).

Step 2: Express diagonal and vectors

Consider \( \overrightarrow{AB} = \vec{u} \) and \( \overrightarrow{AD} = \vec{v} \). In a parallelogram, the diagonals bisect each other, therefore, vector \( \overrightarrow{DB} \) is equal to vector \( \overrightarrow{AC} - \overrightarrow{AB} \).

Step 3: Apply vector addition

Using vector addition properties, we say \( \overrightarrow{DB} = \overrightarrow{CD} + \overrightarrow{CB} = \overrightarrow{AC} - \overrightarrow{AB} \).

Step 4: Use the dot product definition

The dot product \( \overrightarrow{DB} \cdot \overrightarrow{AB} = (\overrightarrow{AC} - \overrightarrow{AB}) \cdot \overrightarrow{AB} = \overrightarrow{AC} \cdot \overrightarrow{AB} - \overrightarrow{AB} \cdot \overrightarrow{AB} \).

Step 5: Use properties of dot product

Given \( \overrightarrow{AB} \cdot \overrightarrow{AB} = |AB|^2 = a^2 \), we also need to find \( \overrightarrow{AC} \cdot \overrightarrow{AB} \). Employ the law of cosines in triangle \( ACB \), we have \( |AC|^2 = |AB|^2 + |BC|^2 - 2|AB||BC|\cos(\angle BAC) \).

Step 6: Use properties of triangle

In triangle \( ABC \), by the law of cosine: \( c^2 = a^2 + b^2 - 2ab \cdot \cos(\angle ACB) \), thus \( \overrightarrow{AC} \cdot \overrightarrow{AB} = ab \cos(\angle ACB) = \frac{c^2 - a^2 - b^2}{2} \).

Step 7: Solve for the dot product

Substitute the results back in: \( \overrightarrow{DB} \cdot \overrightarrow{AB} = \frac{c^2 - a^2 - b^2}{2} - a^2 \), simplifying to \( \frac{3a^2 + b^2 - c^2}{2} \).

Step 8: Choose the correct answer

The correct expression that matches this result is given by option (A): \( \frac{3a^2 + b^2 - c^2}{2} \).

Key Concepts

Vector Dot Product

The dot product is a fundamental concept in vector algebra. It gives a measure of how much one vector goes in the direction of another. In simpler terms, it tells us how two vectors relate in terms of direction and magnitude.

To calculate the dot product of two vectors, we multiply their corresponding components and add the products. This can be represented mathematically as:

• If \( \vec{u} = (u_1, u_2) \) and \( \vec{v} = (v_1, v_2) \), then \( \vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 \).

Another way to approach the dot product is by using magnitudes and the angle between the vectors:

• \( \vec{u} \cdot \vec{v} = |\vec{u}| |\vec{v}| \cos(\theta) \)

where \( \theta \) is the angle between the vectors \( \vec{u} \) and \( \vec{v} \). This formula is especially useful in geometry when working with shapes like parallelograms, where angles between adjacent sides are involved.

Law of Cosines

The Law of Cosines is a critical mathematical tool used to find relationships between the lengths of sides in a triangle. It generalizes the Pythagorean theorem and is essential for solving problems involving non-right-angled triangles.

In a triangle with sides opposite to angles \(A\), \(B\), and \(C\), the Law of Cosines is given by:

• \( c^2 = a^2 + b^2 - 2ab\cos(C) \)

This formula helps in finding a side if you know the other two sides and the included angle, or to find an angle when you know all the sides. It is particularly useful in solving parallelogram problems, as you can apply this law to the triangles formed by the diagonals and sides of the parallelogram.

In our parallelogram problem, we use the Law of Cosines to find the dot product of vectors that share a common vertex. We solve for the component \(ab \cos(C)\), translating it to a more usable form using the lengths of the sides and diagonal.

Parallelogram Properties

Parallelograms are special four-sided figures with opposite sides that are equal and parallel, along with other unique attributes. Understanding these properties is crucial in vector and geometric calculations involving the parallelogram.

Key properties include:

• Opposite sides are equal in length \(( |AB| = |CD| \text{ and } |AD| = |BC| )\).
• Opposite angles are equal \(( \angle A = \angle C \text{ and } \angle B = \angle D ) \).
• Consecutive angles are supplementary \(( \angle A + \angle B = 180^\circ )\).
• The diagonals bisect each other at their point of intersection.

These properties are instrumental when solving for vector quantities using the dot product. For example, the fact that diagonals bisect helps determine the relationships between different vectors formed by the vertices of a parallelogram. Knowing these properties reinforces the reasoning in each step of solving parallelogram problems.