Problem 60 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three unit vectors such that \(\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}\). if \(\lambda=\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}\) and \(\vec{d}=\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}\), then the ordered pair, \((\lambda, \vec{d})\) is equal to: \(\quad\) [Jan. 7, 2020 (II)] (a) \(\left(\frac{3}{2}, 3 \vec{a} \times \vec{c}\right)\) (b) \(\left(-\frac{3}{2}, 3 \vec{c} \times \vec{b}\right)\) (c) \(\left(\frac{3}{2}, 3 \vec{b} \times \vec{c}\right)\) (d) \(\left(-\frac{3}{2}, 3 \vec{a} \times \vec{b}\right)\)

Step-by-Step Solution

Answer

The ordered pair is \((-\frac{3}{2}, 3\vec{a} \times \vec{b})\), which is option (d).

Step 1: Analyze Given Condition for Vectors

We know that three unit vectors \( \vec{a}, \vec{b}, \) and \( \vec{c} \) satisfy the vector addition equation: \( \vec{a} + \vec{b} + \vec{c} = \overrightarrow{0} \). This implies that \( \vec{c} = - (\vec{a} + \vec{b}) \).

Step 2: Compute the Magnitude Equation

Since \( \vec{c} = - (\vec{a} + \vec{b}) \), taking the square magnitude gives us \( ||\vec{c}||^2 = ||-(\vec{a} + \vec{b})||^2 = (\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b}) \). Expand this equation and use the fact that \( \vec{a}, \vec{b}, \) and \( \vec{c} \) are unit vectors: \( 1 = 1 + 1 + 2 (\vec{a} \cdot \vec{b}) \), resulting in the equation \( \vec{a} \cdot \vec{b} = -\frac{1}{2} \).

Step 3: Calculate Lambda (\( \lambda \))

We are given \( \lambda = \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \). Substitute \( \vec{c} = -(\vec{a} + \vec{b}) \) and known dot products: \( \vec{b} \cdot \vec{c} = \vec{b} \cdot (-(\vec{a} + \vec{b})) = - \vec{b} \cdot \vec{a} - 1 = \frac{1}{2} - 1 = -\frac{1}{2} \), \( \vec{c} \cdot \vec{a} = (-(\vec{a} + \vec{b})) \cdot \vec{a} = -1 + \frac{1}{2} = -\frac{1}{2} \). Therefore, \( \lambda = -\frac{1}{2} -\frac{1}{2} -\frac{1}{2} = -\frac{3}{2} \).

Step 4: Find Vector \( \vec{d} \)

We have \( \vec{d} = \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} \). Substitute \( \vec{c} = -(\vec{a} + \vec{b}) \): \( \vec{b} \times \vec{c} = \vec{b} \times (-(\vec{a} + \vec{b})) = -\vec{b} \times \vec{a} \), \( \vec{c} \times \vec{a} = (-(\vec{a} + \vec{b})) \times \vec{a} = - \vec{b} \times \vec{a} \). Hence, \( \vec{d} = \vec{a} \times \vec{b} + (-\vec{b} \times \vec{a}) + (-\vec{b} \times \vec{a}) = 3\vec{a} \times \vec{b} \).

Step 5: Identify the Correct Ordered Pair

From steps 3 and 4, we found \( \lambda = -\frac{3}{2} \) and \( \vec{d} = 3 \vec{a} \times \vec{b} \). The correct ordered pair is therefore (\(-\frac{3}{2}, 3\vec{a} \times \vec{b}\)). This matches option (d) in the given choices.

Key Concepts

Unit Vectors

Unit vectors are vectors with a magnitude of 1. They are used to indicate direction without affecting the length of other vectors in mathematical operations. In calculations, they help to standardize results and simplify problems by focusing mainly on the vector's direction.
When working with unit vectors, it’s easy to manipulate equations, because their magnitude is always 1. This can be particularly useful in problems where you have multiple vectors interacting. For example, in the problem involving vectors \( \vec{a}, \vec{b}, \) and \( \vec{c} \), knowing they are unit vectors simplifies the solution process. You don't need to compute lengths separately, as any calculations naturally respect the unit nature of these vectors.
Some key points to remember about unit vectors:

• Magnitude is 1, expressed mathematically as \( ||\vec{u}|| = 1 \).
• They simplify vector equations by focusing on direction.
• Unit vectors are often represented by a hat symbol (e.g., \( \hat{i}, \hat{j}, \hat{k} \)).

Using unit vectors, you can also construct other vectors by multiplying them with a scalar, thus preserving direction while altering magnitude.

Dot Product

The dot product, also known as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. In vector algebra, it is the product of the magnitudes of two vectors and the cosine of the angle between them.
The formula for the dot product of two vectors \( \vec{a} = (a_1, a_2, a_3) \) and \( \vec{b} = (b_1, b_2, b_3) \) is:\[\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3\]This results in a scalar, which distinguishes it from the cross product. The dot product is particularly useful for finding the angle between two vectors or determining if they are perpendicular (i.e., orthogonal).
A few highlights about the dot product:

• The result is always a scalar.
• If the dot product is zero, the vectors are orthogonal.
• It helps to calculate work done when force and displacement vectors are involved.

In the given problem, the dot product helps determine the value of \( \lambda \), which reflects how much two vectors align in the same or opposite directions.

Cross Product

The cross product, or vector product, of two vectors in three-dimensional space results in a third vector perpendicular to the plane containing the original vectors. This new vector has a magnitude equal to the area of the parallelogram that the vectors span.
Given vectors \( \vec{a} \) and \( \vec{b} \), the cross product is defined as:\[\vec{a} \times \vec{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1)\]The direction of the resulting vector follows the right-hand rule, meaning that if you curl the fingers of your right hand from the first vector to the second, your thumb will point in the direction of the cross product.
Key points about the cross product:

• Result is a vector.
• The magnitude of the cross product is \( |vec{a} \times \vec{b}| = |vec{a}| |vec{b}| \sin(\theta) \), where \( \theta \) is the angle between \( \vec{a} \) and \( \vec{b} \).
• The cross product is zero if the vectors are parallel.

In our problem, finding the cross product is critical for calculating the vector \( \vec{d} \). It shows how the vectors combine to form a direction-resultant vector.

Vector Addition

Vector addition is the process of combining two or more vectors to get a resultant vector. It is one of the simplest operations you can perform on vectors and is often depicted graphically using the head-to-tail method.
To add two vectors, such as \( \vec{a} \) and \( \vec{b} \), you place the tail of \( \vec{b} \) at the head of \( \vec{a} \), and the resultant vector \( \vec{r} \) is drawn from the tail of \( \vec{a} \) to the head of \( \vec{b} \). Algebraically, vector addition is commutative and associative:
The equation for adding vectors is:\[\vec{a} + \vec{b} = (a_1 + b_1, a_2 + b_2, a_3 + b_3)\]Important properties of vector addition include:

• Commutative: \( \vec{a} + \vec{b} = \vec{b} + \vec{a} \).
• Associative: \( (\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c}) \).
• Has an identity element, \( \overrightarrow{0} \), such that \( \vec{a} + \overrightarrow{0} = \vec{a} \).

In the exercise, vector addition helps confirm that the sum of the three vectors \( \vec{a}, \vec{b}, \) and \( \vec{c} \) is zero, enabling calculation simplifications based on the geometric properties of these vectors.