Problem 35 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(b\) and \(c\) are any two non-collinear unit vectors and \(a\) is any vector, then \((a \cdot b) b+(a \cdot c) c+\frac{a \cdot(b+c)}{|b+c|^{2}}(b \times c)=\) (A) \(a\) (B) \(b\) (C) \(c\) (D) none of these

Step-by-Step Solution

Answer

(A) \(a\)

Step 1: Understand Vector Operations

The expression involves dot products, cross products, and a division by the magnitude squared. These are standard vector operations: - The dot product \( a \cdot b \) measures the product of the vector magnitudes and the cosine of the angle between them. - The cross product \( b \times c \) results in a vector perpendicular to both \( b \) and \( c \).- The term \( \frac{a \cdot(b+c)}{|b+c|^{2}} \) is a scalar projection factor.

Step 2: Simplify the Expression

We need to simplify the given equation: \((a \cdot b) b + (a \cdot c) c + \frac{a \cdot (b+c)}{|b+c|^{2}} (b \times c)\)Let's express \(a\) in its components along \(b\) and \(c\) and residual vectors. Notice first that \[u = a - (a \cdot b) b - (a \cdot c) c\]This describes the part of \(a\) that is perpendicular to the plane formed by \(b\) and \(c\).

Step 3: Expression of as Zero

According to the perpendicularity properties of cross products and scalar expressions, \(b \times c\) would add zero effect in the expression along the plane. Hence, introducing the expression for u simplifies to:\[\big((a \cdot b) b + (a \cdot c) c + \frac{a \cdot (b+c)}{|b+c|^{2}} (b \times c)\big)= a\]

Step 4: Check Against Given Choices

The simplified expression lands at representation of the vector a directly in terms of its components. From the simplification evaluations, we established the vector as:- Representation stays as \(a\)\ from calculations matching to expression as \(a = (a \cdot b)b + (a \cdot c)c\) along \(b\) and \(c\). Thus indicated by option.

Step 5: Solution Verification

Testing sections experimentally can verify geometric interpretation holds by looking at characteristics of normal engagement contractions anticipatively alongside engaged elements including \(b\times c\). It's confirmed correctness thereof final optable reliant on \(D)\) adequate observation.

Key Concepts

Dot Product

The dot product is an essential operation in vector algebra and is denoted as \( a \cdot b \). It is a way of multiplying two vectors to produce a scalar. The result tells us something about the vectors' orientation towards each other.

The calculation of the dot product involves:

• Taking the magnitudes (or lengths) of both vectors.
• Multiplying these magnitudes by the cosine of the angle \( \theta \) between them.

The formula can be written as:\[ a \cdot b = |a| |b| \cos(\theta)\]If the dot product is zero, it indicates that the vectors are orthogonal, or perpendicular to each other. This property is useful in various geometric and physical applications, such as determining the angles between directions originated by vectors.

It's also important in projections, as it helps in breaking down a vector along another vector's direction, as used in our exercise to express components along unit vectors \( b \) and \( c \).

Cross Product

The cross product is another fundamental vector operation. Instead of a scalar, the cross product, represented as \( b \times c \), results in another vector. This new vector is perpendicular to the plane formed by the original two vectors.

Here are key points about the cross product:

• The magnitude of the cross product is equal to the area of the parallelogram spanned by the two vectors.
• Mathematically, it's given by: \[ |b \times c| = |b| |c| \sin(\theta) \] where \( \theta \) is the angle between \( b \) and \( c \).
• The direction follows the right-hand rule, meaning if you point your index finger in the direction of \( b \) and your middle finger in the direction of \( c \), your thumb will point in the direction of \( b \times c \).

In the context of the original exercise, since \( b \) and \( c \) are unit vectors and \( b \times c \) is added to a term involving dot products, it highlights perpendicular components that do not contribute to the vector along the plane defined by \( b \) and \( c \).

Unit Vectors

A unit vector is a vector that has a magnitude of exactly 1. It is primarily used to denote direction. When we refer to unit vectors \( b \) and \( c \) in the exercise, we emphasize their solely directional nature.

Here are some notable aspects of unit vectors:

• Used to simplify vector calculations by focusing only on direction.
• Any non-zero vector can be turned into a unit vector by dividing it by its magnitude.

Given a vector \( v \), the unit vector \( \hat{v} \) can be found by:\[\hat{v} = \frac{v}{|v|}\]In our exercise, the presence of unit vectors makes calculations of dot and cross products more straightforward, as the calculations involve simpler components derived from magnitudes that are always 1. Studying operations with unit vectors builds a foundation for more complex manipulations with vectors of different magnitudes.