Problem 74 - A Complete Resource Book in Mathematics for JEE Main 2017

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: (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: Identify the vectors

We are given two vectors: \( \mathbf{A} = \hat{i} - \hat{j} \) and \( \mathbf{B} = \hat{i} + 2\hat{j} \). Our objective is to find a unit vector that is coplanar with these vectors and is also perpendicular to the first vector, \( \mathbf{A} \).

Step 2: Determine the perpendicular vector

To find a vector perpendicular to \( \mathbf{A} = \hat{i} - \hat{j} \), say \( \mathbf{X} = x\hat{i} + y\hat{j} \), ensure their dot product is zero: \( \mathbf{A} \cdot \mathbf{X} = (\hat{i} - \hat{j}) \cdot (x\hat{i} + y\hat{j}) = x - y = 0 \). Thus, \( x = y \).

Step 3: Express vector in terms of y

Using \( x = y \), the expression for \( \mathbf{X} \) becomes \( y\hat{i} + y\hat{j} = y(\hat{i} + \hat{j}) \). This indicates that \( \mathbf{X} \) is a scalar multiple of \( \hat{i} + \hat{j} \).

Step 4: Normalize the vector

Make \( \mathbf{X} \) a unit vector: first, calculate the magnitude of \( \hat{i} + \hat{j} \): \[ ||\mathbf{X}|| = \sqrt{1^2 + 1^2} = \sqrt{2} \].The unit vector is then \( \frac{\hat{i} + \hat{j}}{\sqrt{2}} \).

Step 5: Check coplanarity with second vector

Vector \( \hat{i} + \hat{j} \) is a scalar multiplication of a coplanar vector to \( \hat{i} + 2\hat{j} \) because linear combinations of any two vectors remain in the same plane.

Step 6: Verify from options and conclude

The unit vector we calculated as \( \frac{1}{\sqrt{2}}(\hat{i} + \hat{j}) \) is not matched with options B, C, or D. Option (D) none of these is the answer.

Key Concepts

Dot Product

The dot product is a fundamental operation in vector mathematics. It helps us find the relationship between two vectors in terms of their direction. In simpler terms, the dot product tells us how much one vector extends in the direction of another. To calculate the dot product of two vectors, multiply their corresponding components and sum up the results. If we have vectors \( \mathbf{A} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \) and \( \mathbf{B} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k} \), their dot product \( \mathbf{A} \cdot \mathbf{B} \) is given by: \[ a_1b_1 + a_2b_2 + a_3b_3 \]Some key properties of the dot product include:

• It is commutative: \( \mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A} \).
• If the dot product is zero, the vectors are perpendicular (at a 90-degree angle).

In the exercise, the zero dot product condition is used to ensure two vectors are perpendicular.

Unit Vector

A unit vector is a vector with a magnitude of one. It's used to indicate direction rather than size. In essence, it normalizes a vector to become one unit long, while still pointing in the same direction as the original vector. To find a unit vector \( \mathbf{U} \) in the direction of a vector \( \mathbf{V} \), divide the vector by its magnitude: \[ \mathbf{U} = \frac{\mathbf{V}}{||\mathbf{V}||} \]where \( ||\mathbf{V}|| \) is the magnitude of \( \mathbf{V} \).In the original exercise, the goal was to find a unit vector that is coplanar with given vectors, and perpendicular to one of them. The unit vector we obtain has a magnitude of one, which doesn't match any of the given options in the multiple choice answers.

Coplanar Vectors

Vectors are termed coplanar if they lie within the same plane. In simpler terms, a plane is a flat, two-dimensional surface, and any vector that lies in this plane is called coplanar with others in the same plane.Three or more vectors are coplanar if they can be expressed as linear combinations of two of them. This means that the third vector can be made by adding and scaling (or multiplying by a constant) the first two vectors. In the exercise, it's stated that any linear combination of the given vectors \(\hat{i} - \hat{j}\) and \(\hat{i} + 2\hat{j}\) will result in another vector that lies in the same plane, making it coplanar.

Perpendicular Vectors

Perpendicular vectors intersect at a 90-degree angle. In the realm of physics and mathematics, two vectors are perpendicular if their dot product is zero. This relationship can be represented as follows: if vectors \( \mathbf{A} \) and \( \mathbf{B} \) are perpendicular, then:\[ \mathbf{A} \cdot \mathbf{B} = 0 \]The purpose of the exercise involved finding a vector that was perpendicular to one vector \( \hat{i} - \hat{j} \). By ensuring the dot product with \( \hat{i} - \hat{j} \) was zero, this condition of perpendicularity was satisfied. This plays a critical role in understanding vector norm and directionality in vector mathematics.