Problem 34 - Master Resource Book in JEE Main Physics

Question

If \(\mathbf{P} \cdot \mathbf{Q}=0\), then \(|\mathbf{P} \times \mathbf{Q}|\) is (a) \(|\mathrm{P}||\mathrm{Q}|\) (b) zero (c) 1 (d) \(\sqrt{P Q}\)

Step-by-Step Solution

Answer

The correct answer is (a) |P||Q|.

Step 1: Understand the condition

We are given that \(\mathbf{P} \cdot \mathbf{Q} = 0\). This indicates that the vectors \(\mathbf{P}\) and \(\mathbf{Q}\) are orthogonal to each other.

Step 2: Recall cross product properties

The magnitude of the cross product \(|\mathbf{P} \times \mathbf{Q}|\) is calculated using the formula \(|\mathbf{P}||\mathbf{Q}|\sin \theta|\), where \(\theta\) is the angle between \(\mathbf{P}\) and \(\mathbf{Q}\).

Step 3: Use orthogonality to find \(\sin \theta\)

Since the vectors are orthogonal, \(\theta = \frac{\pi}{2}\), so \(\sin \theta = \sin \frac{\pi}{2} = 1\).

Step 4: Calculate the magnitude of the cross product

Substitute the value of \(\sin \theta\) into the cross product formula: \(|\mathbf{P} \times \mathbf{Q}| = |\mathbf{P}||\mathbf{Q}||\sin \theta|\)Therefore, \(|\mathbf{P} \times \mathbf{Q}| = |\mathbf{P}||\mathbf{Q}|\) since \(\sin \theta = 1\).

Key Concepts

Orthogonal Vectors

In the realm of vectors, two vectors are called orthogonal if they meet at a right angle. Mathematically, this means their dot product equals zero. When you find that \(\mathbf{P} \cdot \mathbf{Q} = 0\), this equation reveals that vectors \(\mathbf{P}\) and \(\mathbf{Q}\) are orthogonal.
The dot product, in this context, is a handy tool to check orthogonality.

• If the dot product is zero, then the vectors are orthogonal.

This property simplifies many calculations in geometry and physics because it directly implies that the vectors are perpendicular to each other.

Cross Product

The cross product of two vectors, represented as \(\mathbf{P} \times \mathbf{Q}\), results in a new vector that is perpendicular to both of the original vectors. The magnitude of the cross product is determined by the formula \(|\mathbf{P} \times \mathbf{Q}| = |\mathbf{P}||\mathbf{Q}|\sin \theta|\), where \(\theta\) is the angle between them.
The direction of the resulting vector follows the right-hand rule. If you point the fingers of your right hand in the direction of \(\mathbf{P}\) and curl them towards \(\mathbf{Q}\), your thumb will point in the direction of \(\mathbf{P} \times \mathbf{Q}\).

• The magnitude is at its maximum when the vectors are orthogonal, as \(\sin \theta = 1\).

This is why the solution to the exercise was \(|\mathbf{P}||\mathbf{Q}|\) when the vectors were orthogonal.

Dot Product

The dot product of two vectors \(\mathbf{P}\) and \(\mathbf{Q}\) is a scalar quantity expressed as \(\mathbf{P} \cdot \mathbf{Q} = |\mathbf{P}||\mathbf{Q}|\cos \theta\). This operation combines both magnitude and direction information from the vectors.
In geometric terms, the dot product expresses how much of one vector goes in the direction of another. For orthogonal vectors, this contribution is zero. Hence, when \(\theta = \frac{\pi}{2}\), the cosine of the angle becomes zero:

• This is the case for orthogonal vectors where \(\mathbf{P} \cdot \mathbf{Q} = 0\).

The dot product helps in understanding the alignment and relationship between vectors.

Angle Between Vectors

The angle between vectors \(\mathbf{P}\) and \(\mathbf{Q}\) can be understood through both the dot product and cross product. It is measured in the range from 0 to \(\pi\). For example, when \(\mathbf{P} \cdot \mathbf{Q} = 0\), the angle \(\theta\) between these vectors is \(\frac{\pi}{2}\), signifying they are orthogonal.
Understanding the angle helps determine the relationship between vectors:

• If \(\theta = 0\), vectors are parallel.
• If \(\theta = \frac{\pi}{2}\), vectors are perpendicular.

Both the cross and dot products make use of this angle to describe how vectors interact, allowing us to derive whether vectors are parallel, perpendicular, or at some other angle.