Problem 52 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

\(\vec{a}, \vec{b}, \vec{c}\) are 3 vectors, such that \(\vec{a}+\vec{b}+\vec{c}=0\), \(|\vec{a}|=1,|\vec{b}|=2,|\vec{c}|=3\), then \(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}\) is equal to \([2003]\) (a) 1 (b) 0 (c) \(-7\) (d) 7

Step-by-Step Solution

Answer

(c) -7

Step 1: Express \\vec{b} and \\vec{c} in terms of \\vec{a}

Given that \( \vec{a} + \vec{b} + \vec{c} = 0 \), we can express \( \vec{b} \) as \( \vec{b} = -\vec{a} - \vec{c} \).

Step 2: Expand Dot Product Formula

We need to find \( \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \), which can be rewritten using \( \vec{b} = -\vec{a} - \vec{c} \) as:\[ \vec{a} \cdot (-\vec{a} - \vec{c}) + (-\vec{a} - \vec{c}) \cdot \vec{c} + \vec{c} \cdot \vec{a}. \]

Step 3: Simplify Each Term

Simplify each term separately:- \( \vec{a} \cdot (-\vec{a} - \vec{c}) = -\vec{a} \cdot \vec{a} - \vec{a} \cdot \vec{c} = -1 - \vec{a} \cdot \vec{c} \).- \( (-\vec{a} - \vec{c}) \cdot \vec{c} = -\vec{a} \cdot \vec{c} - \vec{c} \cdot \vec{c} = -\vec{a} \cdot \vec{c} - 9 \).- \( \vec{c} \cdot \vec{a} = \vec{c} \cdot \vec{a} \).

Step 4: Combine Terms

Combine the simplified terms:\[ -1 - \vec{a} \cdot \vec{c} - \vec{a} \cdot \vec{c} - 9 + \vec{c} \cdot \vec{a} = -10 - 2\vec{a} \cdot \vec{c} + \vec{c} \cdot \vec{a}. \]

Step 5: Relate \\vec{a} \\cdot \\vec{c} and \\vec{c} \\cdot \\vec{a}

Since the dot product is commutative, \( \vec{a} \cdot \vec{c} = \vec{c} \cdot \vec{a} \), set: \( x = \vec{a} \cdot \vec{c} \).Thus, the expression simplifies to: \(-10 - 2x + x = -10 - x \).

Step 6: Find Expression Using Identity

Now, solve for \( \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \):Since \( \vec{a} + \vec{b} + \vec{c} = 0 \):\( \vec{a} - \vec{c} = -\vec{b} \)and use squared terms:\( |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 = 1 + 4 + 9 = 14 \). \( |\vec{a} + \vec{b} + \vec{c}|^2 = 0 \),Using identity:14 + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 0\( 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = -14 \)\( \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = -7 \).

Step 7: Check Calculations and Closure

Verify calculations to ensure accuracy. The solution agrees with given conditions after re-checking math. Final answer is \(-7\).

Key Concepts

Dot Product

The dot product is a fundamental concept in vector algebra. It provides a way to multiply two vectors and get a scalar. To compute the dot product of two vectors, you multiply their corresponding components and then sum those results. Mathematically, for vectors \( \vec{a} = [a_1, a_2, a_3] \) and \( \vec{b} = [b_1, b_2, b_3] \), the dot product is defined as: \( \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3 \).
Another way to express the dot product is through the cosine of the angle \( \theta \) between two vectors: \( \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta) \), where \(|\vec{a}|\) and \(|\vec{b}|\) are the magnitudes of the vectors.
The dot product has several properties:

• Commutative: \( \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} \)
• Distributive: \( \vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} \)
• Scalar Multiplication: \( (c\vec{a}) \cdot \vec{b} = c(\vec{a} \cdot \vec{b}) \)

Understanding the dot product is key for solving problems involving angles between vectors or projections of one vector onto another.

Vector Addition

Vector addition is a simple yet crucial operation in vector algebra, allowing us to combine multiple vectors into a single resultant vector. When adding vectors \( \vec{a} \) and \( \vec{b} \), it involves adding their corresponding components. Mathematically, if \( \vec{a} = [a_1, a_2, a_3] \) and \( \vec{b} = [b_1, b_2, b_3] \), their sum is: \( \vec{a} + \vec{b} = [a_1+b_1, a_2+b_2, a_3+b_3] \).
An important concept related to vector addition is the triangle law. When two vectors are placed head to tail, the sum \( \vec{a} + \vec{b} \) is the vector that extends from the free tail of the first to the free head of the second.
Also relevant is the parallelogram law, which states that if two vectors represent adjacent sides of a parallelogram, their sum is the diagonal of that parallelogram.
Vector addition is both commutative and associative:

• Commutative: \( \vec{a} + \vec{b} = \vec{b} + \vec{a} \)
• Associative: \( (\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c}) \)

These properties simplify calculations involving multiple vector sums. Knowing how to add vectors is essential for resolving vector-related problems, like calculating the net force from several applied forces.

Magnitude of a Vector

The magnitude of a vector, often referred to as its "length" or "norm", quantifies the size of the vector regardless of its direction. Representing the vector \( \vec{a} = [a_1, a_2, a_3] \), its magnitude is calculated using the following formula:
\[ |\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \].
For any vector, its magnitude is always a non-negative value, and it gives an intuitive measurement of the vector's extension in space. The magnitude of a zero vector is always zero, and vectors of equal magnitudes can vary in direction but will still have the same "size" or length.
Magnitudes are significant in a few key applications:

• Normalizing Vectors: Creating unit vectors by dividing by the magnitude, yielding a vector of length 1.
• Distance Calculation: Determining the distance between two points by transforming the position vectors into a single vector.
• Cross Product: Part of calculating the area spanned by two vectors.

Understanding how to calculate and interpret the magnitude is essential for applications in physics and engineering, where both size and direction are vital factors.