Problem 46 - Master Resource Book in JEE Main Physics

Question

Two vectors \(\mathbf{a}\) and \(\mathbf{b}\) are such that \(|\mathbf{a}+\mathbf{b}|=|\mathbf{a}-\mathbf{b}|\) What is the angle between a and \(\mathbf{b}\) ? (a) \(0^{\circ}\) (b) \(90^{\circ}\) (c) \(60^{\circ}\) (d) \(180^{\circ}\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Given Condition

We are given the condition \(|\mathbf{a}+\mathbf{b}|=|\mathbf{a}-\mathbf{b}|\). This means that the magnitude (or length) of the sum of the vectors \(\mathbf{a}\) and \(\mathbf{b}\) is equal to the magnitude of the difference of the vectors \(\mathbf{a}\) and \(\mathbf{b}\).

Step 2: Use the Magnitude Formula

The magnitude of a vector \(\mathbf{v}\) being \(\mathbf{a} + \mathbf{b}\) or \(\mathbf{a} - \mathbf{b}\) is given by \(|\mathbf{v}| = \sqrt{\mathbf{v} \cdot \mathbf{v}}\). For \(\mathbf{a} + \mathbf{b}\), it is \(\sqrt{(\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} + \mathbf{b})}\) and for \(\mathbf{a} - \mathbf{b}\), it is \(\sqrt{(\mathbf{a} - \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b})}\).

Step 3: Expand the Dot Products

Using the distributive property, calculate:\((\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} + \mathbf{b}) = \mathbf{a} \cdot \mathbf{a} + 2\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{b}\) and\((\mathbf{a} - \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}) = \mathbf{a} \cdot \mathbf{a} - 2\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{b}\).

Step 4: Set the Magnitudes Equal

Since \(|\mathbf{a}+\mathbf{b}|=|\mathbf{a}-\mathbf{b}|\), equate the squared forms:\(\mathbf{a} \cdot \mathbf{a} + 2\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{b} = \mathbf{a} \cdot \mathbf{a} - 2\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{b}\).

Step 5: Simplify the Expression

From the equation in Step 4, cancel out terms to get: \(2\mathbf{a} \cdot \mathbf{b} = -2\mathbf{a} \cdot \mathbf{b}\). Therefore, \(4\mathbf{a} \cdot \mathbf{b} = 0\), which simplifies to \(\mathbf{a} \cdot \mathbf{b} = 0\).

Step 6: Interpret the Dot Product Result

Since \(\mathbf{a} \cdot \mathbf{b} = 0\), the vectors \(\mathbf{a}\) and \(\mathbf{b}\) are orthogonal. This means the angle between them is \(90^{\circ}\).

Key Concepts

Vector Magnitude

The magnitude of a vector is like its "length". Imagine you are holding a string stretched out straight - that length gives you a sense of the size of the vector. For any vector \( \mathbf{v} = \langle x, y, z \rangle \), its magnitude \(|\mathbf{v}|\) is found using the formula \(|\mathbf{v}| = \sqrt{x^2 + y^2 + z^2}\). This is an extension of the Pythagorean theorem into higher dimensions, allowing us to calculate the vector's length no matter how many components it has.
Understanding vector magnitudes is essential as it helps compare the sizes of different vectors and is fundamental in physics and engineering. When two vectors, \(\mathbf{a}\) and \(\mathbf{b}\), are combined (either added or subtracted), their resulting magnitude can tell us how these vectors interact or influence each other in a given space.

Dot Product

The dot product is a way to multiply two vectors, resulting in a scalar (a single number) instead of another vector. This operation helps us understand the angle between the vectors and their relative orientation in space. For vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \), the dot product is given by the formula: \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \).
What makes the dot product fascinating is that it ties into the angle between the vectors; specifically, \( \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos(\theta) \), where \(\theta\) is the angle between them. If the dot product is zero, \( \cos(\theta) \) is zero, meaning the vectors are perpendicular. Understanding how the dot product works gives insight into the geometry of vector fields, helping us solve problems in mathematics and physics by providing a concise method to determine alignment or orthogonality of vectors.

Orthogonal Vectors

Orthogonal vectors are vectors that meet at a right angle, or \(90^{\circ}\). When vectors \( \mathbf{a} \) and \( \mathbf{b} \) are orthogonal, their dot product equals zero: \( \mathbf{a} \cdot \mathbf{b} = 0 \). This is an important concept because orthogonality implies that the vectors do not affect each other's magnitude in their respective directions.
In practical terms, finding orthogonal vectors is crucial in various fields such as computer graphics, where independent color channels must remain unaffected by changes in other channels. In physics, forces acting perpendicular to each other can simplify understanding and calculations of movements. Recognizing when vectors are orthogonal helps in optimizing solutions and ensuring clarity in vector-based analyses. Understanding this concept also plays a role in techniques like finding the angle between two vectors, revealing important spatial relationships in vector algebra.