Problem 74 - A Complete Resource Book in Mathematics for JEE Main

Question

Given two vectors are \(\hat{i}-\hat{j}\) and \(\hat{i}+2 \hat{j}\) the unit vector coplanar with the two vectors and perpendicular to first is: (A) \(\frac{\underline{\phantom{xx}}}{\sqrt{t}}(\hat{i}\) \(\hat{j})\) (B) \(\frac{1}{\sqrt{5}}(2 \hat{i}+\hat{j})\) (C) \(\pm \frac{1}{\sqrt{2}}(\hat{i}+\hat{k})\) (D) none of these

Step-by-Step Solution

Answer

(D) None of these

Step 1: Understand the Problem

We need to find a unit vector that is coplanar with the given vectors \( \hat{i}-\hat{j} \) and \( \hat{i}+2\hat{j} \), and is perpendicular to the first vector \( \hat{i}-\hat{j} \).

Step 2: Find the Vector Perpendicular to First

A vector perpendicular to \( \hat{i}-\hat{j} \) should satisfy the dot product condition. Let this vector be \( a \hat{i} + b \hat{j} + c \hat{k} \). We set up the condition \( (\hat{i}-\hat{j}) \cdot (a \hat{i} + b \hat{j}) = 0 \).\[a - b = 0 \implies a = b\\]

Step 3: Solve For the Coplanar Vector Without \(\hat{k}\)

Since this vector is coplanar with \( \hat{i} \) and \( \hat{j} \), and we don't need the \( \hat{k} \) component, we let \( c = 0 \). Therefore, the vector is \( a(\hat{i} + \hat{j}) \).

Step 4: Find a Specific Solution for \(a\) and \(b\)

We already found \( a = b \). So any vector of the form \( a (\hat{i} + \hat{j}) \) will be perpendicular to \( \hat{i} - \hat{j} \). For simplicity, we set \( a = 1 \), giving us a vector \( \hat{i} + \hat{j} \).

Step 5: Normalize to Get the Unit Vector

To make the vector \( \hat{i} + \hat{j} \) a unit vector, we calculate its magnitude:\[\text{Magnitude} = \sqrt{1^2 + 1^2} = \sqrt{2}\]The unit vector is:\[ \frac{1}{\sqrt{2}}(\hat{i} + \hat{j}) \]

Key Concepts

Coplanar Vectors

Vectors are coplanar if they lie within the same plane. In simpler terms, imagine a flat piece of paper where these vectors sit without sticking out into the third dimension. Whenever you deal with vectors in a geometric problem, identifying their coplanar nature helps simplify the calculations.

• For the vectors \( \hat{i}-\hat{j} \) and \( \hat{i}+2\hat{j} \), notice they involve only the \( \hat{i} \) and \( \hat{j} \) components.
• This suggests they lie entirely within the 2D plane formed by these components, assuming no movement along \( \hat{k} \).

This property allows for easy calculations of dot products and other vector operations within the set plane.

Unit Vectors

A unit vector is a vector with a magnitude of exactly 1. The purpose of unit vectors is to denote direction without altering it through magnitude effects. This is extremely important in calculations where only direction matters, such as physics and engineering problems.

• To convert a vector into a unit vector, divide each component of the vector by its magnitude.
• For instance, the vector \( \hat{i} + \hat{j} \) has a magnitude of \( \sqrt{2} \).
• The corresponding unit vector would be \( \frac{1}{\sqrt{2}}(\hat{i} + \hat{j}) \).

This process, known as normalization, ensures the vector retains its direction while standardizing the magnitude to 1.

Dot Product

The dot product is a valuable operation that provides a scalar result from two vectors. It indicates the extent to which two vectors align with each other.

• Mathematically, \( \mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z \).
• In our situation, it helps find a vector perpendicular to \( \hat{i}-\hat{j} \).
• We set the dot product condition \( (\hat{i}-\hat{j}) \cdot (a \hat{i} + b \hat{j}) = 0 \), simplifying it to \( a - b = 0 \).

This tells us how vector components relate when they're perpendicular: they result in a dot product of zero.

Vector Normalization

Vector normalization involves adjusting the vector's magnitude to 1 while maintaining its direction. This is crucial when determining unit vectors.

• The process involves dividing a vector by its magnitude.
• For a vector \( \mathbf{v} = a \hat{i} + b \hat{j} + c \hat{k} \), the magnitude is \( \sqrt{a^2 + b^2 + c^2} \).
• A normalized vector is \( \frac{1}{\text{Magnitude}}(a \hat{i} + b \hat{j} + c \hat{k}) \).

In our example, to normalize \( \hat{i} + \hat{j} \), we calculate its magnitude as \( \sqrt{2} \) and use it to find the unit vector. This step is essential for ensuring vectors are correctly scaled for various applications like physics, graphics, and robotics.