Problem 39 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(\hat{\mathrm{x}}, \hat{\mathrm{y}}\) and \(\hat{\mathrm{z}}\) are three unit vectors in three-dimensional space, then the minimum value of \(|\hat{\mathrm{x}}+\hat{\mathrm{y}}|^{2}+|\hat{\mathrm{y}}+\hat{\mathrm{z}}|^{2}+|\hat{\mathrm{z}}+\hat{\mathrm{x}}|^{2} \quad[\) Online April \(\mathbf{1 2}, \mathbf{2 0 1 4}]\) (a) \(\frac{3}{2}\) (b) 3 (c) \(3 \sqrt{3}\) (d) 6

Step-by-Step Solution

Answer

(b) 3

Step 1: Understand the Problem

We are given three unit vectors, \(\hat{x}, \hat{y}, \hat{z}\), and need to find the minimum value of the expression \(|\hat{x}+\hat{y}|^{2}+|\hat{y}+\hat{z}|^{2}+|\hat{z}+\hat{x}|^{2}\). Each term represents the square of the magnitude of the sum of two unit vectors.

Step 2: Use Vector Algebra Operations

To solve this, use the formula for the magnitude square of a sum of two vectors: \(|\hat{a} + \hat{b}|^2 = (\hat{a} + \hat{b}) \cdot (\hat{a} + \hat{b}) = |\hat{a}|^2 + |\hat{b}|^2 + 2(\hat{a} \cdot \hat{b})\). Since they are unit vectors, \(|\hat{a}|^2 = 1\) and \(|\hat{b}|^2 = 1\), so each term becomes \(2 + 2(\hat{a} \cdot \hat{b})\).

Step 3: Sum All Terms

Calculate the sum of all three expressions: \[|\hat{x}+\hat{y}|^2 + |\hat{y}+\hat{z}|^2 + |\hat{z}+\hat{x}|^2 = (2 + 2(\hat{x} \cdot \hat{y})) + (2 + 2(\hat{y} \cdot \hat{z})) + (2 + 2(\hat{z} \cdot \hat{x}))\]This simplifies to: \[6 + 2((\hat{x} \cdot \hat{y}) + (\hat{y} \cdot \hat{z}) + (\hat{z} \cdot \hat{x}))\]

Step 4: Find the Minimum Value

To minimize the expression, minimize \((\hat{x} \cdot \hat{y}) + (\hat{y} \cdot \hat{z}) + (\hat{z} \cdot \hat{x})\). The minimum value occurs when the sum is as negative as possible. For orthogonal unit vectors, each dot product term can be \(-1\), but this would contradict all vectors being unit vectors. However, setting one dot product to \(-1\) and others to \(0\) achieves a minimum feasible value of \(-1\).

Step 5: Calculate Minimum Case

Assume \(\hat{x} \cdot \hat{y} = -1\), \(\hat{y} \cdot \hat{z} = 0\), \(\hat{z} \cdot \hat{x} = 0\) as the arrangement (others similar), thus the sum is:\[6 + 2(-1 + 0 + 0) = 6 - 2 = 4\]

Step 6: Verify Additional Cases

Check alternate configurations, but they yield values greater than 4 or match it (considering symmetry and mutual angles of vectors as per the constraints of being unit vectors). Thus, 4 is confirmed as the minimum value of the expression.

Key Concepts

Unit Vectors

Unit vectors are fundamental entities in vector algebra, playing a crucial role in describing directions in space without regard to magnitude. Each unit vector has a magnitude, or length, of exactly 1. This simplifies calculations in physics and engineering, where direction without magnitude is often required.
In three-dimensional space, the commonly used unit vectors are typically denoted as \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\), which align with the x-, y-, and z-axes, respectively. When using these in calculations, you can be sure that their magnitude is always 1, allowing you to focus on their orientation.
Understanding unit vectors' key properties simplifies determining the direction of any regular vector. By dividing any vector by its magnitude, you derive a unit vector pointing in the same direction.

Dot Product

The dot product, also known as the scalar product, is a crucial operation for understanding interactions between vectors in vector algebra. It combines two vectors and results in a scalar, reflecting the magnitude of their projection along one another. The formula is as follows:
\[\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos(\theta)\]
where \(\theta\) is the angle between \(\mathbf{a}\) and \(\mathbf{b}\). This means the dot product is a measure of how much one vector extends in the direction of another.
The calculation of the dot product in component form is easily performed as:
\[\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\]
when each vector is expanded in three dimensions (\(a_1, a_2, a_3\)) and (\(b_1, b_2, b_3\)).

• If the dot product is positive, the angle between the vectors is acute.
• If it is zero, the vectors are perpendicular.
• If negative, the angle is obtuse.

Understanding these properties of the dot product illuminates how vectors are individually and collectively oriented, which is vital for operations such as determining if vectors are orthogonal or for evaluating forces and movements in physical systems.

Magnitude of Vectors

The magnitude of a vector is essentially its length and is computed in a straightforward manner using the vector's components. For a vector \(\mathbf{v} = (v_1, v_2, v_3)\) in three-dimensional space, its magnitude \(|\mathbf{v}|\) is calculated as:
\[|\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}\]
This value provides an idea of the vector's "size" irrespective of its direction.
The concept of the magnitude is similar to measuring the hypotenuse in a right triangle. Each component of the vector represents a side, and the magnitude is equivalent to the diagonal length joining these endpoints.

• Magnitude helps determine the "distance" a vector represents.
• It's vital for normalizing vectors, which converts them into unit vectors.
• Used widely in phsyics and engineering to represent quantities such as speed, force, or displacement.

In vector operations, calculating the magnitude is often a preliminary step before further evaluation, revealing how significant an effect a particular vector will have in a given context.