Problem 96 - A Complete Resource Book in Mathematics for JEE Main

Question

The vector \(\vec{a}=\alpha \hat{i}+2 \hat{j}+\beta \hat{k}\) lies in the plane of the vectors \(\vec{b}=\hat{i}+\hat{j}\) and \(\vec{c}=\hat{j}+\hat{k}\) and bisects the angle between \(\vec{b}\) and \(\vec{c}\). Then which one of the following gives possible values of \(\alpha\) and \(\beta ?\) (A) \(\alpha=2, \beta=2\) (B) \(\alpha=1, \beta=2\) (C) \(\alpha=2, \beta=1\) (D) \(\alpha=1, \beta=1\)

Step-by-Step Solution

Answer

\(\alpha = 1, \beta = 1\) (Option D).

Step 1: Define the condition for the vector to be in the plane

A vector \(\vec{a}\) is in the plane spanned by \(\vec{b}\) and \(\vec{c}\) if it can be represented as a linear combination of \(\vec{b}\) and \(\vec{c}\). Thus, \(\vec{a} = p \vec{b} + q \vec{c}\) for some scalars \(p\) and \(q\). First, let's express \(\vec{a}\): \[\vec{a} = \alpha \hat{i} + 2 \hat{j} + \beta \hat{k}\]. Similarly, \(\vec{b} = \hat{i} + \hat{j}\) and \(\vec{c} = \hat{j} + \hat{k}\).

Step 2: Expand the linear combination

We expand the linear combination \(\vec{a} = p \vec{b} + q \vec{c}\):\[ p(\hat{i} + \hat{j}) + q(\hat{j} + \hat{k}) = p\hat{i} + (p+q)\hat{j} + q\hat{k} \].Comparing the components with \(\vec{a} = \alpha \hat{i} + 2 \hat{j} + \beta \hat{k}\), we equate them: \(p = \alpha\), \(p + q = 2\), and \(q = \beta\).

Step 3: Solve the system of equations

We solve the equations: 1. \(p = \alpha\), 2. \(q = \beta\), 3. \(p + q = 2\).Substituting \(\alpha = p\) and \(\beta = q\) into equation 3 gives \(\alpha + \beta = 2\).

Step 4: Define the angle bisector condition

For \(\vec{a}\) to bisect the angle between \(\vec{b}\) and \(\vec{c}\), the direction of \(\vec{a}\) must be equal to the normalized sum of \(\vec{b}\) and \(\vec{c}\). Compute the sum: \(\vec{b} + \vec{c} = \hat{i} + 2\hat{j} + \hat{k}\). This matches \(\vec{a}\) if it is not zero after normalization.

Step 5: Check for matching coefficients

Compare coefficients to find compatible values of \(\alpha\) and \(\beta\) that also satisfy the angle bisector condition. Given \(\alpha + \beta = 2\), trial values: \(\alpha = 1, \beta = 1\) meet both conditions as \(\vec{a} = \vec{b}+\vec{c}\), and it satisfies being equal to sum for angle bisector.

Key Concepts

Linear Combination of Vectors

Understanding linear combination is crucial when dealing with vectors. A vector is said to be a linear combination of other vectors if it can be expressed as a sum of those vectors, each multiplied by a scalar. Imagine it like creating a new recipe using different amounts of your existing ingredients.

For example, if \( \vec{a} = p \vec{b} + q \vec{c}\), then \( \vec{a}\) is a linear combination of \( \vec{b}\) and \( \vec{c}\). Here, the scalars \( p \) and \( q \) represent the "amount" of each vector needed. It's like mixing a certain measure of milk and sugar to get your perfect cup of coffee.

In the exercise, \( \vec{a} = \alpha \hat{i} + 2 \hat{j} + \beta \hat{k}\) is expressed as \( \vec{a} = p (\hat{i} + \hat{j}) + q (\hat{j} + \hat{k})\). By expanding, we get: \( \alpha = p\), \( p + q = 2\), \( \beta = q\). Solving these linear equations helps find specific values of \( \alpha \) and \( \beta \), which ensures \( \vec{a}\) lies in the plane defined by \( \vec{b}\) and \( \vec{c}\).

Hence, mastering linear combinations enables us to represent vectors simply and understand their relationships.

Angle Bisector

An angle bisector is a line or vector that divides an angle into two equal parts. In vector terms, when a vector bisects an angle formed by two other vectors, it points in the direction that cuts the angle into two equal smaller angles.

For the vector \( \vec{a} = \alpha \hat{i} + 2 \hat{j} + \beta \hat{k}\) to bisect the angle between vectors \( \vec{b} = \hat{i} + \hat{j}\) and \( \vec{c} = \hat{j} + \hat{k}\), its direction must be the same as the normalized sum of these two vectors. Think of this as finding the middle path between two vectors, ensuring that you are equidistant to each.

The normalized sum vector can be calculated by adding the vectors \( \vec{b}\) and \( \vec{c}\): \( \vec{b} + \vec{c} = \hat{i} + 2\hat{j} + \hat{k}\). For \( \vec{a} \) to bisect the angle, its direction must be consistent with this calculated direction. The condition is met when \( \alpha = 1\) and \( \beta = 1\), matching the vector \( \vec{b} + \vec{c}\) after checking for normalization.

Vector Equations

Vector equations play a fundamental role in defining and solving problems related to vectors. These equations express relationships between vectors and can be used to find unknown vector components or to verify whether vectors satisfy certain conditions.

In our scenario, the original problem requires solving vector equations to determine the values of \( \alpha \) and \( \beta \). By representing \( \vec{a}\) using \( \vec{b}\) and \( \vec{c}\), we end up forming a system of linear equations:

• \( p = \alpha \)
• \( q = \beta \)
• \( p + q = 2 \)

This system arises from using the conditions of linear combination and the angle bisector concept.

Solving these equations allows us to verify the correct match \( \alpha + \beta = 2 \), which then leads to discovering potential values aligned with the required criteria. Thus vector equations help in systematically breaking down and solving problems involving vector components and their relationships.