Problem 48 - Master Resource Book in JEE Main Physics

Question

If \(\mathbf{A} \cdot \mathbf{B}=0\) and \(\mathbf{A} \times \mathbf{B}=\mathbf{1}\), then \(\mathbf{A}\) and \(\mathbf{B}\) are (a) perpendicular unit vectors (b) parallel unit vectors (c) parallel (d) perpendicular

Step-by-Step Solution

Answer

(a) perpendicular unit vectors

Step 1: Understand Dot Product Condition

The dot product of two vectors, \(\mathbf{A} \cdot \mathbf{B} = 0\), means that \(\mathbf{A}\) and \(\mathbf{B}\) are perpendicular vectors. This product is zero when the angle \(\theta\) between the vectors is 90 degrees, since \(\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos(\theta)\). Thus, \(\cos(\theta) = 0\), implying \(\theta = 90^\circ\).

Step 2: Analyze Cross Product Condition

The cross product \(\mathbf{A} \times \mathbf{B} = \mathbf{1}\) means that the magnitude of the result is 1, and the result vector is in a direction that is perpendicular to both \(\mathbf{A}\) and \(\mathbf{B}\). Since the magnitude of the cross product is given by \(|\mathbf{A}| |\mathbf{B}| \sin(\theta)\), and since \(\mathbf{A} \cdot \mathbf{B} = 0\) implies \(\theta = 90^\circ\), \(\sin(\theta) = 1\). Thus, \(|\mathbf{A}| |\mathbf{B}| = 1\) if both are unit vectors.

Step 3: Combine Conditions

From the dot product condition, we know \(\mathbf{A}\) and \(\mathbf{B}\) are perpendicular. From the cross product condition, both are unit vectors because \(|\mathbf{A}| \cdot |\mathbf{B}| = 1\). Therefore, \(\mathbf{A}\) and \(\mathbf{B}\) are perpendicular unit vectors.

Key Concepts

Dot Product

When we talk about the dot product in vector algebra, we're referring to a way of multiplying two vectors that gives us a scalar—just a single number. For two vectors \( \mathbf{A} \) and \( \mathbf{B} \), the dot product is written as \( \mathbf{A} \cdot \mathbf{B} \). This is calculated using the formula:

• \( \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos(\theta) \),

where \( |\mathbf{A}| \) and \( |\mathbf{B}| \) are the magnitudes of the vectors, and \( \theta \) is the angle between them.
When the dot product is zero, like in our case \( \mathbf{A} \cdot \mathbf{B} = 0 \), it indicates the vectors are perpendicular.
Perpendicular means they form a 90-degree angle with each other, making the cosine of the angle zero, as \( \cos(90^\circ) = 0 \).
This concept helps in understanding how vectors relate to each other in terms of direction and orientation.

Cross Product

The cross product is another way to multiply two vectors, but unlike the dot product, it results in a new vector. This new vector is perpendicular to both of the original vectors involved in the multiplication. For vectors \( \mathbf{A} \) and \( \mathbf{B} \), their cross product is written as \( \mathbf{A} \times \mathbf{B} \).
The magnitude of this vector is calculated as:

• \( |\mathbf{A} \times \mathbf{B}| = |\mathbf{A}| |\mathbf{B}| \sin(\theta) \),

where \( \sin(\theta) \) depends on the angle \( \theta \) between \( \mathbf{A} \) and \( \mathbf{B} \).
In our scenario, \( \mathbf{A} \times \mathbf{B} = \mathbf{1} \) signifies that the resulting vector has a magnitude of 1 and is perpendicular to both vectors.
This condition generally confirms that \( \mathbf{A} \) and \( \mathbf{B} \) are unit vectors (i.e., their magnitude equals one) because \( |\mathbf{A}| \cdot |\mathbf{B}| = 1 \).
This leads to the logical conclusion that they are indeed perpendicular unit vectors.

Unit Vectors

Unit vectors are fundamental in vector algebra as they provide direction without considering magnitude. Converting a vector into a unit vector involves dividing the vector by its magnitude, resulting in a vector with a magnitude of exactly one.
A unit vector in the direction of \( \mathbf{A} \) is expressed as \( \hat{\mathbf{A}} = \frac{\mathbf{A}}{|\mathbf{A}|} \).
Unit vectors are essential because they often simplify calculations in physics and engineering. They define directions clearly without scaling by length, which is beneficial in breaking down complex vector operations.
In the given problem, since both \( \mathbf{A} \) and \( \mathbf{B} \) are determined to be unit vectors, we know that:

• \( |\mathbf{A}| = 1 \) and \( |\mathbf{B}| = 1 \).

This simplifies understanding of their interaction through both dot and cross products, emphasizing how they form perpendicular unit vectors whose interactions are straightforward and model many real-world applications.