Problem 55 - Master Resource Book in JEE Main Physics

Question

Three vectors \(\mathbf{A}, \mathbf{B}\) and \(\mathbf{C}\) satisfy the relation \(\mathbf{A B}=0\) and \(\mathbf{A C}=0 .\) If \(\mathbf{B}\) and \(\mathbf{C}\) are not lying in the same plane, then \(\mathbf{A}\) is parallel to (a) \(\mathrm{B}\) (b) \(\mathrm{C}\) (c) \(\mathrm{B} \times \mathrm{C}\) (d) \(\mathrm{BC}\)

Step-by-Step Solution

Answer

\( \mathbf{A} \) is parallel to \( \mathbf{B} \times \mathbf{C} \).

Step 1: Understanding the Problem

We are given that vector \( \mathbf{A} \) is perpendicular to both vectors \( \mathbf{B} \) and \( \mathbf{C} \). This is indicated by the dot products \( \mathbf{A} \cdot \mathbf{B} = 0 \) and \( \mathbf{A} \cdot \mathbf{C} = 0 \). The vectors \( \mathbf{B} \) and \( \mathbf{C} \) do not lie in the same plane, which means they are linearly independent.

Step 2: Analyzing Dot Product Implications

If \( \mathbf{A} \cdot \mathbf{B} = 0 \) and \( \mathbf{A} \cdot \mathbf{C} = 0 \), \( \mathbf{A} \) must be orthogonal to both \( \mathbf{B} \) and \( \mathbf{C} \). This non-collinearity and the fact that \( \mathbf{B} \) and \( \mathbf{C} \) are not coplanar suggest that \( \mathbf{A} \) lies along the line perpendicular to the plane formed by \( \mathbf{B} \) and \( \mathbf{C} \).

Step 3: Vectors and Plane Formulation

In a three-dimensional space, two vectors form a plane. The vector orthogonal to both \( \mathbf{B} \) and \( \mathbf{C} \) is given by the cross product \( \mathbf{B} \times \mathbf{C} \). Since \( \mathbf{A} \) is not in the plane of \( \mathbf{B} \) and \( \mathbf{C} \), it must be parallel to their cross product.

Step 4: Conclusion

Because \( \mathbf{A} \) is perpendicular to both \( \mathbf{B} \) and \( \mathbf{C} \), and \( \mathbf{B} \) and \( \mathbf{C} \) are not coplanar, \( \mathbf{A} \) must be parallel to the vector \( \mathbf{B} \times \mathbf{C} \), the one that is orthogonal to the plane formed by \( \mathbf{B} \) and \( \mathbf{C} \). Thus, the correct choice is (c) \( \mathrm{B} \times \mathrm{C} \).

Key Concepts

Dot Product

The dot product is a way to multiply two vectors and results in a scalar. It's particularly useful for determining the angle between two vectors or finding if they are perpendicular. When the dot product of two vectors is zero, it indicates that the vectors are orthogonal. The formula for the dot product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is given by:

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

where \( \theta \) is the angle between the vectors.
If \( \cos\theta \) is zero, then \( \theta \) must be \(90^\circ\), indicating the vectors are perpendicular.
In this exercise, since \( \mathbf{A} \cdot \mathbf{B} = 0 \) and \( \mathbf{A} \cdot \mathbf{C} = 0 \), \( \mathbf{A} \) is perpendicular to both \( \mathbf{B} \) and \( \mathbf{C} \). This is crucial for understanding vector relationships in three-dimensional space.

Cross Product

The cross product results in a new vector that is orthogonal, or perpendicular, to the original two vectors. The cross product \( \mathbf{B} \times \mathbf{C} \) is especially important when dealing with three-dimensional vector spaces.
Unlike the dot product, the result of a cross product is a vector, not a scalar.
The magnitude of the cross product follows:

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

Here, \( \theta \) represents the angle between vectors \( \mathbf{B} \) and \( \mathbf{C} \).
This vector is perpendicular to the plane formed by \( \mathbf{B} \) and \( \mathbf{C} \). In our context, since \( \mathbf{A} \) is parallel to the cross product \( \mathbf{B} \times \mathbf{C} \), it means \( \mathbf{A} \) is orthogonal to the plane which \( \mathbf{B} \) and \( \mathbf{C} \) form.

Orthogonality

Orthogonality is a concept that describes a situation where two vectors are perpendicular to each other.
To determine orthogonality, the dot product between the two vectors must be zero. This is a crucial condition and is applied in various areas, such as determining perpendicular directions in geometry or in optimizing equations in mathematics.
In our scenario, since \( \mathbf{A} \cdot \mathbf{B} = 0 \) and \( \mathbf{A} \cdot \mathbf{C} = 0 \), \( \mathbf{A} \) is orthogonal to both \( \mathbf{B} \) and \( \mathbf{C} \).
Recognizing this helps in understanding why \( \mathbf{A} \) must lie along the direction normal to the plane created by \( \mathbf{B} \) and \( \mathbf{C} \).

Linear Independence

Linear independence refers to a set of vectors where no vector in the set is a linear combination of the others.
If two vectors are not in the same plane, they are linearly independent, meaning one vector cannot be expressed as a scalar multiple of the other.
In vector algebra, this implies that these vectors span a plane in three-dimensional space. When \( \mathbf{B} \) and \( \mathbf{C} \) are linearly independent, they define a unique plane.
For this exercise, the linear independence of \( \mathbf{B} \) and \( \mathbf{C} \) ensures the existence of a non-zero vector \( \mathbf{B} \times \mathbf{C} \) that is perpendicular to this plane, aligning perfectly with the perpendicular vector \( \mathbf{A} \).
Understanding linear independence assists in distinguishing between vectors in the same plane and those spanning or orthogonal to a plane.