Problem 31 - Master Resource Book in JEE Main Physics

Question

Given, vector, \(\mathbf{A}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) and vector \(\mathbf{B}=3 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}\), then which one of the following statements is true? (a) \(A\) is perpendicular to \(B\) (b) \(A\) is parallel to \(B\) (c) Magnitude of \(A\) is half of that of \(B\) (d) Magnitude of \(B\) is equal to that of \(A\)

Step-by-Step Solution

Answer

Vector \(A\) is parallel to vector \(B\).

Step 1: Calculate the dot product to check for perpendicularity

The first step is to calculate the dot product of vectors \(\mathbf{A}\) and \(\mathbf{B}\). If \(\mathbf{A} \cdot \mathbf{B} = 0\), then the vectors are perpendicular. The dot product is calculated as: \[ \mathbf{A} \cdot \mathbf{B} = (1)(3) + (-1)(-3) + (2)(6) \]Calculate the expression: \[\mathbf{A} \cdot \mathbf{B} = 3 + 3 + 12 = 18\]Since 18 is not equal to 0, \(\mathbf{A}\) is not perpendicular to \(\mathbf{B}\).

Step 2: Check for parallelism using vector coefficients

For vectors to be parallel, they must be scalar multiples of each other. Compare the ratios of coefficients of \(\hat{\mathbf{i}}\), \(\hat{\mathbf{j}}\), and \(\hat{\mathbf{k}}\):\[\frac{3}{1} = 3, \quad \frac{-3}{-1} = 3, \quad \frac{6}{2} = 3\]All ratios are equal, indicating that \(\mathbf{A}\) and \(\mathbf{B}\) are parallel.

Step 3: Calculate magnitudes to check magnitude-based options

Calculate the magnitudes of \(\mathbf{A}\) and \(\mathbf{B}\):Magnitude of \(\mathbf{A}\): \[|\mathbf{A}| = \sqrt{ 1^2 + (-1)^2 + 2^2 } = \sqrt{1 + 1 + 4} = \sqrt{6}\]Magnitude of \(\mathbf{B}\):\[|\mathbf{B}| = \sqrt{ 3^2 + (-3)^2 + 6^2 } = \sqrt{9 + 9 + 36} = \sqrt{54} = 3\sqrt{6}\]The magnitude of \(\mathbf{B}\) is not equal to that of \(\mathbf{A}\) and the magnitude of \(\mathbf{A}\) is \(\frac{1}{3}\) of \(\mathbf{B}\), not \(\frac{1}{2}\). Thus, options (c) and (d) are incorrect.

Key Concepts

Dot Product

The dot product, often called the scalar product, is a fundamental concept in vector algebra. It provides a way of multiplying two vectors, yielding a scalar quantity rather than another vector. This scalar result is crucial for determining the angle between vectors or finding whether they are orthogonal. To calculate the dot product of two vectors \(\mathbf{A} = a_1 \hat{\mathbf{i}} + a_2 \hat{\mathbf{j}} + a_3 \hat{\mathbf{k}}\) and \(\mathbf{B} = b_1 \hat{\mathbf{i}} + b_2 \hat{\mathbf{j}} + b_3 \hat{\mathbf{k}}\), use the formula:\[ \mathbf{A} \cdot \mathbf{B} = a_1b_1 + a_2b_2 + a_3b_3\]

• If the dot product is zero, \(\mathbf{A}\) and \(\mathbf{B}\) are perpendicular.
• If the dot product is positive, the angle between them is less than 90 degrees.
• If it's negative, the angle is more than 90 degrees.

It's important because it helps us understand the relationship between two vectors just by looking at their components in different dimensions.

Parallel Vectors

Two vectors are considered parallel if they are scalar multiples of each other. Simply put, this means that one vector can be obtained by multiplying the other vector by some scalar number. For vectors \(\mathbf{A}\) and \(\mathbf{B}\), they are parallel if their corresponding components have the same ratio. Mathematically, this is expressed as:\[\frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3}\]

• Parallel vectors have the same direction but not necessarily the same magnitude.
• Despite having different lengths, the direction stays consistent.

Understanding parallelism is valuable because it helps simplify vector comparisons, and is necessary for many applications in physics and engineering.

Magnitude of a Vector

The magnitude of a vector, often referred to as the length of the vector, is a measure of its size. It is calculated using the Pythagorean theorem for vectors with components. For a vector \(\mathbf{A} = a_1 \hat{\mathbf{i}} + a_2 \hat{\mathbf{j}} + a_3 \hat{\mathbf{k}}\), the magnitude is calculated as:\[|\mathbf{A}| = \sqrt{a_1^2 + a_2^2 + a_3^2}\]

• The magnitude represents the distance of the vector from the origin in three-dimensional space.
• A zero magnitude means the vector has no length, essentially being a point at the origin.
• Knowing the magnitude is crucial for normalizing vectors, which is the process of scaling them to a unit length.

Magnitudes are especially important in physics and engineering when calculating force, velocity, or any other quantity that has both direction and size.