Problem 26 - Master Resource Book in JEE Main Physics

Question

If the resultant of the vectors \((\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}),(\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}})\) and \(\mathbf{C}\) is a unit vector along the \(\mathrm{y}\)-direction, then \(\mathbf{C}\) is (a) \(-2 \hat{\mathrm{i}}-\hat{\mathrm{k}}\) (b) \(-2 \hat{\mathrm{i}}+\hat{\mathrm{k}}\) (c) \(2 \hat{\mathbf{i}}-\hat{\mathbf{k}}\) (d) \(-2 \hat{\mathrm{i}}+\hat{\mathrm{k}}\)

Step-by-Step Solution

Answer

\( \mathbf{C} = -2 \hat{\mathbf{i}} - \hat{\mathbf{k}} \) is correct, matching option (a).

Step 1: Write the Given Vectors

The problem provides us with two vectors: \( \mathbf{A} = \hat{\mathbf{i}} + 2 \hat{\mathbf{j}} - \hat{\mathbf{k}} \) and \( \mathbf{B} = \hat{\mathbf{i}} - \hat{\mathbf{j}} + 2 \hat{\mathbf{k}} \). A third vector \( \mathbf{C} = a\hat{\mathbf{i}} + b\hat{\mathbf{j}} + c\hat{\mathbf{k}} \) needs to be determined.

Step 2: Write the Condition for the Resultant

The vectors \( \mathbf{A} \), \( \mathbf{B} \), and \( \mathbf{C} \) should sum up to give a resultant vector that is a unit vector in the \( \hat{\mathbf{j}} \) direction, i.e., \( \mathbf{R} = 0 \hat{\mathbf{i}} + 1 \hat{\mathbf{j}} + 0 \hat{\mathbf{k}} \).

Step 3: Express the Sum of the Vectors

The resultant vector \( \mathbf{R} \) is the sum of \( \mathbf{A} \), \( \mathbf{B} \), and \( \mathbf{C} \):\[\mathbf{R} = ( \hat{\mathbf{i}} + 2\hat{\mathbf{j}} - \hat{\mathbf{k}} ) + ( \hat{\mathbf{i}} - \hat{\mathbf{j}} + 2\hat{\mathbf{k}} ) + ( a\hat{\mathbf{i}} + b\hat{\mathbf{j}} + c\hat{\mathbf{k}} ).\]

Step 4: Combine Like Terms

Combine the components of the vectors:- \( \hat{\mathbf{i}} \) terms: \( 1 + 1 + a = 2 + a \)- \( \hat{\mathbf{j}} \) terms: \( 2 - 1 + b = 1 + b \)- \( \hat{\mathbf{k}} \) terms: \( -1 + 2 + c = 1 + c \)So, the resultant vector becomes:\[ ( 2 + a ) \hat{\mathbf{i}} + ( 1 + b ) \hat{\mathbf{j}} + ( 1 + c ) \hat{\mathbf{k}}.\]

Step 5: Set Up Equations for Unit Vector Condition

Since \( \mathbf{R} \) must equal \( 0 \hat{\mathbf{i}} + 1 \hat{\mathbf{j}} + 0 \hat{\mathbf{k}} \), it leads to the equations:- For \( \hat{\mathbf{i}} \): \( 2 + a = 0 \)- For \( \hat{\mathbf{j}} \): \( 1 + b = 1 \)- For \( \hat{\mathbf{k}} \): \( 1 + c = 0 \)

Step 6: Solve the Equations

Solving these equations:- From \( 2 + a = 0 \), we get \( a = -2 \).- From \( 1 + b = 1 \), we get \( b = 0 \).- From \( 1 + c = 0 \), we get \( c = -1 \).Thus, \( \mathbf{C} = -2 \hat{\mathbf{i}} + 0 \hat{\mathbf{j}} - \hat{\mathbf{k}} \), which simplifies to \( -2 \hat{\mathbf{i}} - \hat{\mathbf{k}} \).

Step 7: Match with Options

Comparing the calculated \( \mathbf{C} = -2 \hat{\mathbf{i}} - \hat{\mathbf{k}} \) with the given options, we find that this matches option (a).

Key Concepts

Resultant Vector

In physics, understanding the idea of a *resultant vector* is crucial for analyzing the effects of different forces or movements combined together. A resultant vector is simply the vector obtained when two or more vectors are added together. Think of it as a single vector that represents the cumulative effect of all the individual vectors.

The key points to remember about resultant vectors are:

• It simplifies the problem by reducing multiple vectors into one.
• It has the same effect as all the original vectors combined.
• Calculating the resultant involves vector addition, which will be discussed later.

In the exercise, vectors \( \mathbf{A} \), \( \mathbf{B} \), and \( \mathbf{C} \) combine to create a resultant vector \( \mathbf{R} \), which is a unit vector in the \( \hat{\mathbf{j}} \) direction. This means that after combining all vectors, the net effect is exactly like a unit vector pointing upward along the y-axis. By setting the resultant vector equal to \( 0 \hat{\mathbf{i}} + 1 \hat{\mathbf{j}} + 0 \hat{\mathbf{k}} \), you ensure that the resultant is directed and scaled appropriately.

Unit Vector

A *unit vector* is a vector that has a magnitude (length) of exactly one unit. It is often used to indicate direction without representing any magnitude.

Key characteristics of unit vectors include:

• They are typically used to denote direction in physics problems.
• The unit vector along any coordinate axis must have a magnitude of one.
• Unit vectors are denoted by a circumflex or a hat, such as \( \hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}} \).

In the context of the exercise, the requirement for the resultant vector to be a unit vector in the \( \hat{\mathbf{j}} \) direction informs us that the combined effect of vectors \( \mathbf{A} \), \( \mathbf{B} \), and \( \mathbf{C} \) has no i or k components, only a j component with a magnitude of 1. This specific unit vector indicates a precise direction and magnitude, which helps to steer the calculation of any unknown components in the additive vector, \( \mathbf{C} \).

Vector Addition

*Vector addition* involves combining two or more vectors to obtain a resultant vector. It's a fundamental concept in vector analysis used in physics to understand how different vectors interact with one another.

Here are some essentials to remember about vector addition:

• Vectors are added component-wise; that means combining their i, j, and k components separately.
• The resultant's direction and magnitude depend on the way vectors are summed up in terms of their components.
• Graphically, vectors can be added using the tail-to-head method or parallelogram method.

In the exercise, the process of vector addition is used to sum up vectors \( \mathbf{A} \), \( \mathbf{B} \), and \( \mathbf{C} \) in order to reach a specific resultant. By breaking the problem down, you sum up respective components, which allows you to control and adjust them until you meet the given conditions. Mastering vector addition is crucial for solving problems involving multiple forces or motions, aiding in the understanding of how these influences come together to produce a net effect.