Problem 51 - Master Resource Book in JEE Main Physics

Question

Given that, \(\mathbf{A}+\mathbf{B}+\mathbf{C}=0 .\) Out of three vectors, two are equal in magnitude and the magnitude of third vector is \(\sqrt{2}\) times that of either of the two having equal magnitude. Then, the angles between vectors are given by (a) \(45^{\circ}, 45^{\circ}, 90^{\circ}\) (b) \(90^{\circ}, 135^{\circ}, 135^{\circ}\) (c) \(30^{\circ}, 60^{\circ}, 90^{\circ}\) (d) \(45^{\circ}, 60^{\circ}, 90^{\circ}\)

Step-by-Step Solution

Answer

(b) The angles are \(90^{\circ}, 135^{\circ}, 135^{\circ}\).

Step 1: Express Vectors in Terms of Magnitude

Let's express the vectors \( \mathbf{A}, \mathbf{B}, \mathbf{C} \) in terms of their magnitudes. Assume the magnitudes of \( \mathbf{A} \) and \( \mathbf{B} \) are equal, i.e., \( |\mathbf{A}| = |\mathbf{B}| = a \). The magnitude of \( \mathbf{C} \) will be \( \sqrt{2}a \) as given in the problem.

Step 2: Use the Given Vector Equation

We know that \( \mathbf{A} + \mathbf{B} + \mathbf{C} = 0 \). This implies that \( \mathbf{C} = - (\mathbf{A} + \mathbf{B}) \). So, \( \mathbf{C} \) is the negative sum of \( \mathbf{A} \) and \( \mathbf{B} \).

Step 3: Apply Triangle Properties in Vector Addition

Vectors \( \mathbf{A}, \mathbf{B}, \) and \( \mathbf{C} \) can form a triangle because their sum is zero. For a triangle with sides \( a, a, \sqrt{2}a \), it resembles an isosceles right triangle (by the properties of \( 1, 1, \sqrt{2} \) triangle related to the Pythagorean theorem).

Step 4: Calculate the Vector Angles

In an isosceles right triangle, two angles are \( 45^{\circ} \), and the included angle is \( 90^{\circ} \). So, the angles between \( \mathbf{A}, \mathbf{B}, \mathbf{C} \) are \( 90^{\circ}, 135^{\circ}, 135^{\circ} \), taking into account the opposite orientations due to vector addition.

Key Concepts

Understanding Vector Magnitudes

A vector's magnitude refers to its length or size, and is a crucial component in vector mathematics. It's often represented as \(|\mathbf{A}|\) for vector \(\mathbf{A}\). In our exercise, we deal with three vectors – \(\mathbf{A}, \mathbf{B},\) and \(\mathbf{C}\). Two of these vectors, \(\mathbf{A}\) and \(\mathbf{B}\), have equal magnitudes denoted by \(a\). The third vector, \(\mathbf{C}\), is given to be \(\sqrt{2}\) times this magnitude, or \(\sqrt{2}a\).
Understanding this scalar multiplication is vital, as it tells us how much longer the third vector is compared to the other two. This directly informs the geometrical relationships and angles between these vectors, making vector magnitudes a fundamental aspect of analyzing vector addition problems.

Triangle Properties in Vectors

When you add vectors and their sum equals zero, they can form a closed shape such as a triangle. In our problem, \(\mathbf{A} + \mathbf{B} + \mathbf{C} = 0\) tells us that these vectors form a geometric triangle.
The triangle formed by vectors where the sides are in the ratio of \(1, 1, \sqrt{2}\) indicates an isosceles right triangle due to its specific side lengths. This type of triangle has special properties, such as:

• Two sides are equal, exemplified by \(\mathbf{A}\) and \(\mathbf{B}\).
• It includes a right angle (\(90^{\circ}\)).
• A unique configuration that influences the resultant angles, which are crucial for determining relationships between vectors.

These properties not only help in calculating the angles but also in understanding how vectors interact with each other in space.

Angles Between Vectors

Angles between vectors are calculated based on their directional relationships. In this exercise, because \(\mathbf{A}, \mathbf{B},\) and \(\mathbf{C}\) form an isosceles right triangle, they have specific angle properties.
The included angle in an isosceles right triangle is \(90^{\circ}\), which is the angle between the vector equivalent of the hypotenuse and the other two vectors. Additionally, due to the orientation in negative space (since \(\mathbf{C} = - (\mathbf{A} + \mathbf{B})\)), the angles are adjusted to be \(90^{\circ}, 135^{\circ}, \text{and } 135^{\circ}\).
These angles reflect both their perpendicular and supplementary nature in vector addition. Knowing how to compute and interpret these angles is essential for solving vector equations, allowing you to predict how forces or directions might interact in real-world applications.