Problem 64 - Master Resource Book in JEE Main Physics

Question

If the vectors \(\mathbf{A}=2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}\) and \(\mathbf{B}=5 \hat{\mathbf{i}}-p \hat{\mathbf{j}}\) are parallel to each other, the magnitude of \(\mathbf{B}\) is (a) \(5 \sqrt{5}\) (b) 10 (c) 15 (d) \(2 \sqrt{5}\)

Step-by-Step Solution

Answer

The magnitude of \(\mathbf{B}\) is \(5\sqrt{5}\) (option a).

Step 1: Understanding Parallel Vectors

Two vectors are parallel if one is a scalar multiple of the other. This implies that the ratio of their corresponding components should be equal. For the vectors \(\mathbf{A} = 2 \hat{\mathbf{i}} + 4 \hat{\mathbf{j}}\) and \(\mathbf{B} = 5 \hat{\mathbf{i}} - p \hat{\mathbf{j}}\), they are parallel if \(\frac{2}{5} = \frac{4}{-p}\).

Step 2: Solve for Unknown p

Set the ratio equation \(\frac{2}{5} = \frac{4}{-p}\) and solve for \(p\). Cross-multiply to get \(2(-p) = 5 \times 4\). Simplifying this gives us: \(-2p = 20\) which leads to \(p = -10\).

Step 3: Calculate the Magnitude of \(\mathbf{B}\)

Now with \(p = -10\), the vector \(\mathbf{B}\) becomes \(5 \hat{\mathbf{i}} + 10 \hat{\mathbf{j}}\). The magnitude of \(\mathbf{B}\) is \(\sqrt{(5)^2 + (10)^2}\). Calculate this value: \(\sqrt{25 + 100} = \sqrt{125} = 5\sqrt{5}\).

Step 4: Identify the Correct Option

Compare the calculated magnitude \(5\sqrt{5}\) against the given options: (a) \(5 \sqrt{5}\), (b) 10, (c) 15, (d) \(2 \sqrt{5}\). The correct option is (a) \(5 \sqrt{5}\).

Key Concepts

Parallel Vectors

Vectors are said to be parallel if one vector is a scalar multiple of the other. This essentially means that they have the same or opposite direction in a two-dimensional or three-dimensional space.
Parallel vectors maintain a constant ratio of their corresponding components. For example, if you have two vectors \( \mathbf{A} = a_1 \hat{\mathbf{i}} + a_2 \hat{\mathbf{j}} \) and \( \mathbf{B} = b_1 \hat{\mathbf{i}} + b_2 \hat{\mathbf{j}} \), they are parallel if \( \frac{a_1}{b_1} = \frac{a_2}{b_2} \).

In this problem, determining parallelism helps us set up the equation \( \frac{2}{5} = \frac{4}{-p} \) for vectors \( \mathbf{A} \) and \( \mathbf{B} \). This ratio means vector components align in a straight line, either extending or retracting along the same direction.

Vector Magnitude

The magnitude of a vector is a measure of its length or size, often indicating how intense or substantial it is. In mathematical terms, you can calculate the magnitude of a vector \( \mathbf{V} = x \hat{\mathbf{i}} + y \hat{\mathbf{j}} \) using the Pythagorean theorem as \( \sqrt{x^2 + y^2} \).

For vector \( \mathbf{B} = 5 \hat{\mathbf{i}} + 10 \hat{\mathbf{j}} \), its magnitude is calculated by substituting in the values, resulting in the expression \( \sqrt{(5)^2 + (10)^2} = \sqrt{125} \). With a calculation yielding \( 5\sqrt{5} \), it provides the exact size of the vector in coordinate space. This quantifiable measure allows us to compare its magnitude with other options or vectors.

Solving Equations

Solving equations, especially when dealing with vectors, often involves concepts such as cross-multiplying to isolate variables and find unknowns. When given the equation \( \frac{2}{5} = \frac{4}{-p} \), it's crucial to solve for \( p \) by cross-multiplying, leading to \( 2(-p) = 5 \times 4 \), and then simplifying to \(-2p = 20\).

In this context, rearranging the equation gives us \( p = -10 \). Having found this value, it enables us to verify the orientation or magnitude when plugged back into the vector \( \mathbf{B} \). This approach of solving such equations is a fundamental skill in vector calculations, guiding us to gain answers through algebraic manipulation.