Problem 25 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Consider points \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and \(\mathrm{D}\) with position vectors \(7 \hat{i}-4 \hat{j}+7 \hat{k}, \hat{i}-6 \hat{j}+10 \hat{k},-\hat{i}-3 \hat{j}+4 \hat{k}\) and \(5 \hat{i}-\hat{j}+5 \hat{k}\) respectively. Then \(\mathrm{ABCD}\) is a (a) parallelogram but not a rhombus (b) square (c) rhombus (d) rectangle.

Step-by-Step Solution

Answer

ABCD is a parallelogram, not a rhombus.

Step 1: Calculate Vectors AB, BC, AD

To find if ABCD is a specific type of quadrilateral, calculate vectors \( \vec{AB} \), \( \vec{BC} \), and \( \vec{AD} \) using the position vectors given. \[ \vec{AB} = \vec{B} - \vec{A} = (1 - 7)\hat{i} + (-6 + 4)\hat{j} + (10 - 7)\hat{k} = -6\hat{i} - 2\hat{j} + 3\hat{k} \]\[ \vec{BC} = \vec{C} - \vec{B} = (-1 - 1)\hat{i} + (-3 + 6)\hat{j} + (4 - 10)\hat{k} = -2\hat{i} + 3\hat{j} - 6\hat{k} \]\[ \vec{AD} = \vec{D} - \vec{A} = (5 - 7)\hat{i} + (-1 + 4)\hat{j} + (5 - 7)\hat{k} = -2\hat{i} + 3\hat{j} - 2\hat{k} \]

Step 2: Check for Parallelism

To confirm parallelism (or opposite sides equal in vector magnitude), compare vectors \( \vec{AB} \) and \( \vec{CD} \), \( \vec{BC} \) and \( \vec{AD} \)...\[ \vec{CD} = \vec{D} - \vec{C} = (5 + 1)\hat{i} + (-1 + 3)\hat{j} + (5 - 4)\hat{k} = 6\hat{i} + 2\hat{j} + 1\hat{k} \]Since \( \vec{AB} eq -\vec{CD} \), \( \vec{AB} \) is not parallel to \( \vec{CD} \). Next, calculate the magnitudes.

Step 3: Validate Magnitudes for Rhombus or Other Shape

Calculate the magnitudes to check for a rhombus, rectangle or square:- Magnitude of \(\vec{AB}\): \[ \sqrt{(-6)^2 + (-2)^2 + 3^2} = \sqrt{36 + 4 + 9} = \sqrt{49} = 7 \]- Magnitude of \(\vec{BC}\): \[ \sqrt{(-2)^2 + 3^2 + (-6)^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7 \]- Magnitude of \(\vec{AD}\): \[ \sqrt{(-2)^2 + 3^2 + (-2)^2} = \sqrt{4 + 9 + 4} = \sqrt{17} \]Notice that \(\vec{AD}\) is not equal to \(\vec{AB}\) or \(\vec{BC}\), so it is not a rhombus.

Step 4: Conclusion Based on Observations

Thus, given that opposite vectors are not parallel and not all sides have equal magnitude, \(\mathrm{ABCD}\) can only be a parallelogram.

Key Concepts

Position Vectors

Position vectors are fundamental components in vector mathematics. They provide a way to represent the position of a point in space relative to a fixed origin. In this exercise, position vectors are used to denote the coordinates of points \(A\), \(B\), \(C\), and \(D\). Each vector has components along the axes \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\), which relate to the x, y, and z coordinates respectively.

For example, point \(A\) has the position vector \(7 \hat{i} - 4 \hat{j} + 7 \hat{k}\), meaning it is 7 units along the x-axis, -4 units along the y-axis, and 7 units along the z-axis.

When calculating vectors between points, like \(\vec{AB}\), it involves subtracting the components of vector \(A\) from vector \(B\). This is done for all three axes: \((B_x - A_x) \hat{i}, (B_y - A_y) \hat{j}, (B_z - A_z) \hat{k}\). These calculations convert position vectors into direction vectors that tell us how to move from one point to another in space.

Parallelogram Identification

In geometry, identifying the specific type of quadrilateral can be crucial for solving many mathematical problems. A parallelogram is a four-sided figure with opposite sides that are parallel and equal in length. To identify it using vectors, we must check for parallelism and equal magnitudes of the opposite sides.

Using the vectors derived from the position vectors, such as \(\vec{AB}\) and \(\vec{CD}\), we can determine if they are parallel by checking their directions. Vectors are parallel if one is a scalar multiple of the other. Additionally, we verify that the magnitudes of the opposite vectors are equal using the distance formula.

In this example, not all conditions are met as \(\vec{AB}\) is not parallel to \(\vec{CD}\), leading to the conclusion that the quadrilateral \(\mathrm{ABCD}\) is a general parallelogram but not a rhombus or rectangle.

Magnitude of Vectors

The magnitude of a vector is akin to the length of the vector in a geometrical sense. It is calculated using the Pythagorean theorem in three dimensions. For any vector \(\vec{v} = a\hat{i} + b\hat{j} + c\hat{k}\), the magnitude \(|\vec{v}|\) is given by \(\sqrt{a^2 + b^2 + c^2}\).

In this solution, we calculated magnitudes for vectors \(\vec{AB}\), \(\vec{BC}\), and \(\vec{AD}\). Each calculation involved squaring the vector's components, summing them up, and then taking the square root.

For example, the magnitude of \(\vec{AB}\) is \(\sqrt{(-6)^2 + (-2)^2 + 3^2} = 7\). The magnitudes help determine the type of quadrilateral; in this case, since not all vectors have equal magnitudes, the shape cannot be a rhombus or a square. The magnitude is essential in understanding vector lengths and hence the distances between points in space.