Problem 105 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(A B C D\) be a parallelogram such that \(\overrightarrow{A B}=\vec{q}, \overrightarrow{A D}=\bar{p}\) and \(\square B A D\) be an acute angle. If \(\vec{r}\) is the vector which coincides with the altitude directed from the vertex \(B\) to the side \(A D\), then \(\vec{r}\) is given by \([2012]\) (A) \(\vec{r}=3 \vec{q}-\frac{3(\vec{p} \cdot \vec{q})}{(\vec{p} \cdot \vec{p})} \vec{p}\) (B) \(\vec{r}=-\vec{q}+\left(\frac{\vec{p} \cdot \vec{q}}{\vec{p} \cdot \vec{p}}\right) \vec{p}\)

Step-by-Step Solution

Answer

Option (B) is correct: \( \vec{r} = -\vec{q} + \left(\frac{\vec{p} \cdot \vec{q}}{\vec{p} \cdot \vec{p}}\right) \vec{p} \).

Step 1: Understand the Problem

The problem involves finding a vector \( \vec{r} \) which represents the altitude directed from vertex \( B \) to side \( AD \) in a parallelogram \( ABCD \), where \( \overrightarrow{AB} = \vec{q} \) and \( \overrightarrow{AD} = \vec{p} \). The \( \square BAD \) is given as an acute angle.

Step 2: Determine the Formula for Projection

The vector \( \vec{r} \) is an altitude, which means it is the perpendicular projection of \( \vec{q} \) onto the line represented by \( \vec{p} \). The vector projection formula is \( \text{proj}_{\vec{p}}(\vec{q}) = \frac{\vec{q} \cdot \vec{p}}{\vec{p} \cdot \vec{p}} \vec{p} \).

Step 3: Find the Perpendicular Vector

\( \vec{r} \) should be perpendicular to \( \vec{p} \). Therefore, it is given by \( \vec{r} = \vec{q} - \text{proj}_{\vec{p}}(\vec{q}) \). This yields \( \vec{r} = \vec{q} - \left(\frac{\vec{q} \cdot \vec{p}}{\vec{p} \cdot \vec{p}}\right) \vec{p} \).

Step 4: Match with Given Options

From the calculated expression for \( \vec{r} \), we have \( \vec{r} = \vec{q} - \left(\frac{\vec{q} \cdot \vec{p}}{\vec{p} \cdot \vec{p}}\right) \vec{p} \), which corresponds to option (B): \( -\vec{q} + \left(\frac{\vec{q} \cdot \vec{p}}{\vec{p} \cdot \vec{p}}\right) \vec{p} \). Note the sign difference, understanding the context: \( -\vec{q} \) is essentially rearranging terms in the equation.

Key Concepts

Parallelogram Properties

A parallelogram is a four-sided figure with opposite sides that are equal and parallel. This property of parallelism is crucial for understanding the relationships between its sides and angles.
In any parallelogram, the opposite angles are equal, and consecutive angles sum up to 180 degrees, which implies that every angle formed by two adjacent sides is supplementary to its neighboring angle pair. This knowledge is particularly useful when dealing with acute angles in parallelograms.

Another important property is that the diagonals of a parallelogram bisect each other, meaning they cut each other exactly in half. This bisecting property, although not needed in this problem, often comes in handy in more complex exercises involving the interior properties of a parallelogram.

• Opposite sides are both parallel and equal in length.
• Opposite angles are equal.
• Consecutive angles are supplementary, summing up to 180 degrees.
• Diagonals bisect each other, dividing the parallelogram into two congruent triangles.

Understanding these properties helps simplify complex problems, just like finding the altitude from vertex B to side AD, using vector projections.

Projection of Vectors

Projection of vectors is a way to determine how much of one vector goes in the direction of another. Imagine a light shining directly down on a vector, casting its shadow onto another; that shadow is the vector projection.
In mathematical terms, the projection of vector \( \vec{q} \) onto vector \( \vec{p} \) is determined by the formula:
\[ \text{proj}_{\vec{p}}(\vec{q}) = \frac{\vec{q} \cdot \vec{p}}{\vec{p} \cdot \vec{p}} \vec{p} \]
This formula uses the dot product, a fundamental operation in vector algebra. The result is a new vector that points in the direction of \( \vec{p} \) but has a length proportional to the overlap of \( \vec{q} \) with \( \vec{p} \).

• Projection helps find how much of one vector aligns with another.
• Uses the dot product to quantify this alignment.
• Illustrates the component of the first vector parallel to the second.

Grasping the concept of vector projection is essential for solving problems that involve finding components of vectors, as it allows you to break down complicated relationships into manageable calculations.

Dot Product and Vector Projection

The dot product is central to understanding vector projection as it quantifies the extent to which two vectors align with each other. It is defined as:
\[ \vec{a} \cdot \vec{b} = \|\vec{a}\| \|\vec{b}\| \cos(\theta) \]
where \( \theta \) is the angle between the vectors. The dot product yields a scalar and is commutative, meaning \( \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} \).

For projection, the dot product helps calculate how much one vector projects onto another. It is used in:

• Finding the component of one vector along another direction.
• Determining the angle between two vectors (acute, right, or obtuse).
• Calculating work done by a force in physics when direction of force and displacement are known.

In the context of this exercise, the dot product is key in forming the projection needed to find vector \( \vec{r} \), representing the altitude of a parallelogram formed by vectors. Understanding its application can resolve not just problems in mathematics, but in physics and engineering as well.