Problem 18 - Master Resource Book in JEE Main Physics

Question

The area of the parallelogram represented by the vectors, \(\mathbf{A}=4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}\) and \(\mathbf{B}=2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}\) is (a) 14 units (b) \(7.5\) units (c) 10 units (d) 5 units

Step-by-Step Solution

Answer

The area of the parallelogram is 10 units, corresponding to option (c).

Step 1: Find the Cross Product Formula for Vectors

The area of a parallelogram formed by vectors \( \mathbf{A} \) and \( \mathbf{B} \) is given by the magnitude of their cross product: \( |\mathbf{A} \times \mathbf{B}| \). Understanding this will help in computing the area.

Step 2: Set Up the Cross Product

The vectors given are \( \mathbf{A} = 4 \hat{\mathbf{i}} + 3 \hat{\mathbf{j}} \) and \( \mathbf{B} = 2 \hat{\mathbf{i}} + 4 \hat{\mathbf{j}} \). The cross product \( \mathbf{A} \times \mathbf{B} \) can be represented as a determinant: \( \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \ 4 & 3 & 0 \ 2 & 4 & 0 \end{vmatrix} \).

Step 3: Calculate the Determinant for Cross Product

Evaluate the determinant to find \( \mathbf{A} \times \mathbf{B} \):\( \mathbf{A} \times \mathbf{B} = \hat{\mathbf{i}}(3 \times 0 - 4 \times 0) - \hat{\mathbf{j}}(4 \times 0 - 2 \times 0) + \hat{\mathbf{k}}(4 \times 4 - 3 \times 2) = 0\hat{\mathbf{i}} - 0\hat{\mathbf{j}} + 10\hat{\mathbf{k}} \).This simplifies to \( 10\hat{\mathbf{k}} \).

Step 4: Compute the Magnitude of the Cross Product

The magnitude of the cross product \( |\mathbf{A} \times \mathbf{B}| \) is the area of the parallelogram:\( |\mathbf{A} \times \mathbf{B}| = \sqrt{0^2 + 0^2 + 10^2} = \sqrt{100} = 10 \).

Step 5: Select the Correct Option

The computed area of the parallelogram is 10 square units. Therefore, the correct answer choice is (c) 10 units.

Key Concepts

Parallelogram Area Calculation

Calculating the area of a parallelogram formed by two vectors involves using the vector cross product. The area of a parallelogram given vectors \( \mathbf{A} \) and \( \mathbf{B} \) is equal to the magnitude of their cross product: \( |\mathbf{A} \times \mathbf{B}| \). This formula helps determine how much two vectors "spread out" in space when forming a parallelogram.

In simpler terms, when you have two vectors, they're a bit like sticks in space. The parallelogram they form when connected is a two-dimensional shape. To find out how large this shape is, you calculate the area using their cross product. Mathematically, this cross product generates another vector whose length is exactly the area of this parallelogram.

Visualizing can greatly assist in understanding this concept. Imagine the vectors as lines on a grid, showing the height and base of a parallelogram you are trying to calculate the area for.

Magnitude of Vectors

The magnitude of a vector is a measure of its length or size. It is the equivalent of finding the distance from the origin of a vector to its endpoint in the vector space. For a vector \( \mathbf{V} \) in 3D space, represented as \( \langle x, y, z \rangle \), the magnitude is calculated using the formula: \( |\mathbf{V}| = \sqrt{x^2 + y^2 + z^2} \).

This concept is crucial when dealing with vector operations, such as cross products. In the context of finding the area of a parallelogram, the magnitude is used to derive the final numerical value for this area. It conveys information about the size of the resulting vector from operations like the cross product.

Magnitude tells you how much 'space' a vector takes up, or how great the influence of the vector is in its direction. Just like in everyday measurements, vector magnitude provides a scale for comparing lengths.

Determinants in Vector Calculations

Determinants are mathematical constructs that are used extensively in vector calculations, particularly with regards to cross products. When calculating the cross product of two vectors in 3D, the determinant of a matrix represents these vectors in the basis of \( \hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}} \).

The determinant approach simplifies finding the cross product. For vectors \( \mathbf{A} = 4 \hat{\mathbf{i}} + 3 \hat{\mathbf{j}} \) and \( \mathbf{B} = 2 \hat{\mathbf{i}} + 4 \hat{\mathbf{j}} \), the determinant is set up like this:

• \( \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \ 4 & 3 & 0 \ 2 & 4 & 0 \end{vmatrix} \)

Evaluating this determinant helps obtain the components of the cross product vector. It's like a shortcut that brings together all parts of the vectors into one calculation, providing a broader picture at a glance.

Determinants are not just limited to straightforward vector calculations. They're a fundamental part of linear algebra and are used in solving systems of equations, transforming coordinates, and more. Learning how to calculate and use them will significantly enhance your ability to work with vectors.