Problem 43 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(A B C D\) be a parallelogram such that \(\overrightarrow{A B}=\vec{q}, \overline{A D}=\vec{p}\) and \(\angle B A D\) be an acute angle. If \(\vec{r}\) is the vector that coincide with the altitude directed from the vertex \(\mathrm{B}\) to the side \(A D\), then \(\vec{r}\) is given by : (a) \(\vec{r}=3 \vec{q}-\frac{3(\vec{p} \cdot \vec{q})}{(\vec{p} \cdot \vec{p})} \vec{p}\) (b) \(\vec{r}=-\vec{q}+\frac{(\vec{p} \cdot \vec{q})}{(\vec{p} \cdot \vec{p})} \vec{p}\) (c) \(\vec{r}=\vec{q}-\frac{(\vec{p} \cdot \vec{q})}{(\vec{p} \cdot \vec{p})} \vec{p}\) (d) \(\vec{r}=-3 \vec{q}-\frac{3(\vec{p} \cdot \vec{q})}{(\vec{p} \cdot \vec{p})} \vec{p}\)

Step-by-Step Solution

Answer

The vector \( \vec{r} \) is given by option (c): \( \vec{r} = \vec{q} - \frac{(\vec{p} \cdot \vec{q})}{(\vec{p} \cdot \vec{p})} \vec{p} \).

Step 1: Understanding the Vector Components

We start with the vector \( \vec{q} \), which represents side \( AB \), and the vector \( \vec{p} \), which represents side \( AD \) of the parallelogram. We need to find \( \vec{r} \), the vector representing the altitude from vertex \( B \) to side \( AD \). This altitude is perpendicular to \( \vec{p} \).

Step 2: Finding the Projection of \( \vec{q} \) on \( \vec{p} \)

The component of \( \vec{q} \) along \( \vec{p} \) can be found using the formula for the vector projection: \[ \text{Proj}_{\vec{p}} \vec{q} = \frac{(\vec{p} \cdot \vec{q})}{(\vec{p} \cdot \vec{p})} \vec{p} \] This gives us the part of \( \vec{q} \) that is parallel to \( \vec{p} \).

Step 3: Determining the Perpendicular Component

The vector that is perpendicular to \( \vec{p} \), which is our altitude, \( \vec{r} \), can be found by subtracting the projection from \( \vec{q} \): \[ \vec{r} = \vec{q} - \text{Proj}_{\vec{p}} \vec{q} = \vec{q} - \frac{(\vec{p} \cdot \vec{q})}{(\vec{p} \cdot \vec{p})} \vec{p} \]

Step 4: Choosing the Correct Option

We compare our expression for \( \vec{r} \) from Step 3 with the given options:- (a) \( 3 \vec{q} - \frac{3(\vec{p} \cdot \vec{q})}{(\vec{p} \cdot \vec{p})} \vec{p} \)- (b) \( -\vec{q} + \frac{(\vec{p} \cdot \vec{q})}{(\vec{p} \cdot \vec{p})} \vec{p} \)- (c) \( \vec{q} - \frac{(\vec{p} \cdot \vec{q})}{(\vec{p} \cdot \vec{p})} \vec{p} \)- (d) \( -3 \vec{q} - \frac{3(\vec{p} \cdot \vec{q})}{(\vec{p} \cdot \vec{p})} \vec{p} \)The correct choice that matches our expression for \( \vec{r} \) is (c).

Key Concepts

Parallelogram

A parallelogram is a four-sided shape with opposite sides that are equal and parallel. Each pair of opposite sides are parallel, meaning they are the same distance apart and will never meet. Parallelograms have several key properties that are useful in geometry, particularly in vector analysis.

In the context of vectors, a parallelogram can help visualize the sum or difference of two vectors. The sides of a parallelogram can be represented by vectors. For example, if \( \overrightarrow{AB} = \vec{q} \) and \( \overrightarrow{AD} = \vec{p} \), vectors can help determine properties like angles, altitudes, and projections within the parallelogram.

• The diagonals of a parallelogram bisect each other.
• The opposite angles are equal.
• Each pair of opposite sides are of equal length.

These properties form the foundation for understanding how to manipulate and use vectors to solve geometric problems involving parallelograms.

Projection of vectors

Projection of vectors is a crucial concept in geometry, which helps decompose vectors in terms of their components. When dealing with vectors, you might need to find how much of one vector goes in the direction of another. This is called the projection.

The formula for projecting a vector \(\vec{q}\) onto another vector \(\vec{p}\) is:\[ \text{Proj}_{\vec{p}} \vec{q} = \frac{(\vec{p} \cdot \vec{q})}{(\vec{p} \cdot \vec{p})} \vec{p} \]

This operation gives a vector that points in the same direction as \(\vec{p}\) but has a length that reflects how much \(\vec{q}\) "fits" along \(\vec{p}\).

• The dot product \( \vec{p} \cdot \vec{q} \) is used here to find the magnitude of the projection.
• The denominator \( \vec{p} \cdot \vec{p} \) normalizes \(|\vec{p}|\) to ensure the direction is preserved.

Projection is especially useful in finding components of vectors in specified directions and is fundamental in problems involving force and motion.

Vector components

Vector components break down a vector into parts that are easier to analyze or interpret. When you resolve a vector into components, it allows you to see how much of that vector acts in certain directions.

Think of a vector as an arrow in the plane or space. It has both a magnitude (length) and direction. To understand a vector completely, especially in geometry, you might want to express it in terms of its components along certain lines or axes.

• Vectors can be divided into two types of components: parallel and perpendicular.
• The parallel component follows a specified line (or vector), while the perpendicular component is orthogonal to it.
• Getting the complete picture of a vector's behavior involves knowing both the parallel and perpendicular components.

Being able to decompose a vector into these components underlies much of vector geometry and is a useful technique across many areas of physics and math.

Perpendicular vectors

Perpendicular vectors are vectors that meet at a right angle. This concept is particularly relevant when analyzing geometric figures or determining vector components.

In geometry, two vectors \(\vec{a} \) and \(\vec{b} \) are perpendicular if their dot product is zero: \[ \vec{a} \cdot \vec{b} = 0 \]

This property makes perpendicular vectors extremely valuable because they can simplify calculations and can easily be separated into an independent basis for spaces.

In the exercise, vector \( \vec{r} \) was perpendicular to \( \vec{p} \), representing an altitude. Calculating the perpendicular component involves recognizing or calculating the dot product and utilizing vector projection to isolate the part of the vector that does not align with another. This ensures vectors operate independently along their respective axes, preserving direction.

• Knowing when vectors are perpendicular helps solve complex geometric problems efficiently.
• Perpendicular vectors can represent directions like height in a geometrical figure, providing insight into spatial relationships.

Whether dealing with actual geometric figures or abstract spaces, understanding perpendicular vectors is foundational in vector mathematics.