Problem 35 - Master Resource Book in JEE Main Physics

Question

Given, \(\mathbf{c}=\mathbf{a} \times \mathbf{b}\). The angle which a makes with \(\mathbf{c}\) is (a) \(0^{\circ}\) (b) \(45^{\circ}\) (c) \(90^{\circ}\) (d) \(180^{\circ}\)

Step-by-Step Solution

Answer

The angle between \( \mathbf{a} \) and \( \mathbf{c} \) is \( 90^{\circ} \). (c) is correct.

Step 1: Understanding the Cross Product

The cross product of two vectors, \( \mathbf{a} \times \mathbf{b} \), results in a vector \( \mathbf{c} \) that is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \). This is because the cross product is defined such that the resulting vector is orthogonal to the plane containing \( \mathbf{a} \) and \( \mathbf{b} \).

Step 2: Identifying the Angle

Since \( \mathbf{c} \) is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \), the angle between \( \mathbf{a} \) and \( \mathbf{c} \) is \( 90^{\circ} \). In other words, \( \mathbf{a} \) and \( \mathbf{c} \) are orthogonal, which means they form a right angle.

Step 3: Conclusion

Based on the properties of the cross product, the correct answer is that the angle between \( \mathbf{a} \) and \( \mathbf{c} \) is \( 90^{\circ} \), which corresponds to option (c).

Key Concepts

Orthogonal Vectors

Orthogonal vectors are an essential concept in vector mathematics. When we say that two vectors are orthogonal, we mean that they meet at a right angle, or 90 degrees. This is a special property because it means the vectors do not share any directional components.

• The cross product of two vectors always results in a vector that is orthogonal to the two original vectors.
• For example, if we have vectors \( \mathbf{a} \) and \( \mathbf{b} \), their cross product \( \mathbf{c} = \mathbf{a} \times \mathbf{b} \) is orthogonal to both \( \mathbf{a} \) and \( \mathbf{b} \).

The orthogonality of vectors is an important characteristic. It is widely used in geometry, physics, and engineering to simplify complex vector problems. Understanding this property helps in analyzing the relationship between different directions in space.

Angle Between Vectors

Understanding the angle between vectors is crucial when dealing with vector operations. The angle gives insight into how two vectors are oriented relative to each other.

• The angle between two vectors can be calculated using their dot product or cross product.
• When two vectors are perpendicular, or orthogonal, the angle between them is exactly 90 degrees.

When you compute the cross product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) to get a vector \( \mathbf{c} \), \( \mathbf{c} \) is orthogonal to \( \mathbf{a} \) and \( \mathbf{b} \). Consequently, the angle between \( \mathbf{a} \) and \( \mathbf{c} \) is \( 90^{\circ} \). This underscores the property that the cross product vector is perpendicular to the plane formed by the initial vectors.

Properties of Vector Operations

Vector operations have distinct properties that make them powerful mathematical tools. Understanding these properties is critical for solving vector problems efficiently.

• Distributive Law: Cross product is distributive over vector addition. For vectors \( \mathbf{u}, \mathbf{v}, \) and \( \mathbf{w} \), \( \mathbf{u} \times (\mathbf{v} + \mathbf{w}) = (\mathbf{u} \times \mathbf{v}) + (\mathbf{u} \times \mathbf{w}) \).
• Anticommutative Property: The cross product is anticommutative, meaning \( \mathbf{a} \times \mathbf{b} = - (\mathbf{b} \times \mathbf{a}) \).
• Scalar Multiplication: When a vector is multiplied by a scalar, it scales the resulting vector by that scalar. If \( \mathbf{a} \) is scaled by \( k \), then \( k \mathbf{a} \times \mathbf{b} = k (\mathbf{a} \times \mathbf{b}) \).

These properties simplify the manipulation of vectors and are at the core of many physics and engineering calculations. They help in predicting and explaining various phenomena by understanding how vectors interact with each other.