Problem 29 - Master Resource Book in JEE Main Physics

Question

What vector must be added to the sum of two vectors \(2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) and \(3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}\) so that the resultant is a unit vector along \(Z\)-axis? (a) \(5 \hat{\mathbf{i}}+\hat{\mathrm{k}}\) (b) \(-5 \hat{i}+3 \hat{j}\) (c) \(3 \hat{j}+5 \hat{k}\) (d) \(-3 \hat{j}+2 \hat{\mathrm{k}}\)

Step-by-Step Solution

Answer

The vector is option (b), \(-5 \hat{i} + 3 \hat{j}\).

Step 1: Add the given vectors

First, add the two vectors \( \mathbf{A} = 2 \hat{\mathbf{i}} - \hat{\mathbf{j}} + 3 \hat{\mathbf{k}} \) and \( \mathbf{B} = 3 \hat{\mathbf{i}} - 2 \hat{\mathbf{j}} - 2 \hat{\mathbf{k}} \).\\[ \mathbf{A} + \mathbf{B} = (2 \hat{\mathbf{i}} + 3 \hat{\mathbf{i}}) + (-\hat{\mathbf{j}} - 2 \hat{\mathbf{j}}) + (3 \hat{\mathbf{k}} - 2 \hat{\mathbf{k}}) \]\\[ = 5 \hat{\mathbf{i}} - 3 \hat{\mathbf{j}} + \hat{\mathbf{k}} \]

Step 2: Define the resultant vector

We need to find a vector that, when added to \( \mathbf{A} + \mathbf{B} = 5 \hat{\mathbf{i}} - 3 \hat{\mathbf{j}} + \hat{\mathbf{k}} \), results in a unit vector along the \( Z \)-axis, \( \hat{\mathbf{k}} \). Let this vector be \( \mathbf{C} = x \hat{\mathbf{i}} + y \hat{\mathbf{j}} + z \hat{\mathbf{k}} \). Then the equation becomes:\\[ 5 \hat{\mathbf{i}} - 3 \hat{\mathbf{j}} + \hat{\mathbf{k}} + x \hat{\mathbf{i}} + y \hat{\mathbf{j}} + z \hat{\mathbf{k}} = \hat{\mathbf{k}} \]

Step 3: Equate components

For the resultant to equal \( \hat{\mathbf{k}} \), equate each component:- \( x \hat{\mathbf{i}}: 5 + x = 0 \)- \( y \hat{\mathbf{j}}: -3 + y = 0 \)- \( z \hat{\mathbf{k}}: 1 + z = 1 \)

Step 4: Solve the equations

Solve the equations for \( x \), \( y \), and \( z \):- From \( 5 + x = 0 \), we get \( x = -5 \).- From \( -3 + y = 0 \), we get \( y = 3 \).- From \( 1 + z = 1 \), we get \( z = 0 \).

Step 5: Formulate the solution vector

The vector \( \mathbf{C} \) that must be added is \(-5 \hat{\mathbf{i}} + 3 \hat{\mathbf{j}} + 0 \hat{\mathbf{k}} \). Therefore, the vector is \(-5 \hat{\mathbf{i}} + 3 \hat{\mathbf{j}} \).

Step 6: Verify the solution

Verify that adding \(-5 \hat{\mathbf{i}} + 3 \hat{\mathbf{j}} \) to the sum \(5 \hat{\mathbf{i}} - 3 \hat{\mathbf{j}} + \hat{\mathbf{k}}\) results in \(\hat{\mathbf{k}}\): \\[ (5\hat{\mathbf{i}} - 3\hat{\mathbf{j}} + \hat{\mathbf{k}}) + (-5 \hat{\mathbf{i}} + 3 \hat{\mathbf{j}}) = \hat{\mathbf{k}} \]\The solution is consistent with the requirements.

Key Concepts

Unit Vector

A unit vector is a vector that has a magnitude of exactly one. It is used to indicate a direction without specifying any particular scale. Unit vectors are often represented with a "hat" symbol, such as \( \hat{\mathbf{i}} \), \( \hat{\mathbf{j}} \), and \( \hat{\mathbf{k}} \), which point in the direction of the x, y, and z axes, respectively.

To find if a vector is a unit vector, we calculate its magnitude. In three-dimensional space, the magnitude of a vector \( \mathbf{v} = x\hat{\mathbf{i}} + y\hat{\mathbf{j}} + z\hat{\mathbf{k}} \) is found using the formula: \[ \|\mathbf{v}\| = \sqrt{x^2 + y^2 + z^2} \]If \( \|\mathbf{v}\| = 1 \), then \( \mathbf{v} \) is a unit vector.

In this exercise, the goal is to find a resultant vector that is a unit vector specifically along the Z-axis, represented by \( \hat{\mathbf{k}} \). It means the resultant vector should point in the direction of the Z-axis and have a magnitude of one.

Resultant Vector

The resultant vector is what you obtain when two or more vectors are added together. It represents the cumulative effect of those vectors.

To find the resultant vector, we add up all the individual components of the vectors involved. For example, if we have vectors \( \mathbf{A} = 2\hat{\mathbf{i}} - \hat{\mathbf{j}} + 3\hat{\mathbf{k}} \) and \( \mathbf{B} = 3\hat{\mathbf{i}} - 2\hat{\mathbf{j}} - 2\hat{\mathbf{k}} \), their resultant \( \mathbf{R} \) would be calculated by adding respective components:

• \( i \text{-components}: 2 + 3 = 5 \)
• \( j \text{-components}: -1 - 2 = -3 \)
• \( k \text{-components}: 3 - 2 = 1 \)

Thus, \( \mathbf{R} = 5\hat{\mathbf{i}} - 3\hat{\mathbf{j}} + \hat{\mathbf{k}} \).

In this problem, after finding \( \mathbf{R} \), we need to determine another vector \( \mathbf{C} \) that, when added to this resultant, gives us a unit vector in the Z-direction, \( \hat{\mathbf{k}} \).

Coordinate Vectors

Coordinate vectors are vectors that point in the direction of the coordinate axes and are commonly used to express vectors in a Cartesian coordinate system. They are denoted as \( \hat{\mathbf{i}}, \hat{\mathbf{j}}, \) and \( \hat{\mathbf{k}} \), corresponding to the x, y, and z axes.

When expressing other vectors in terms of these coordinate vectors, components of each vector give you a clear direction and magnitude along each axis. For instance, the vector \( \mathbf{A} = 2\hat{\mathbf{i}} - \hat{\mathbf{j}} + 3\hat{\mathbf{k}} \) means it has:

• 2 units in the positive x-direction,
• -1 unit in the negative y-direction,
• 3 units in the positive z-direction.

This breakdown is powerful because it lets us easily perform operations like addition or projection on vectors by working with their components separately.

To solve problems involving vectors like the one here, it's essential to manage these coordinated directions, add them neatly, and then adjust for any required resultant, as done in the vector addition to reach a unit vector in the Z-axis.