Question

Which one of the following figures represents the vector field \(\mathbf{A} = y\hat{i}\)?
(\(\hat{i}\) is the unit vector along the x-direction)

• A.

  

• B.

  

• C.

  

• D.

  

Hint:

When analyzing a vector field plot, systematically check three properties: 1. \textbf{Direction}: In which way do the arrows point in different regions (e.g., different quadrants)? 2. \textbf{Magnitude}: How does the length of the arrows change as you move around the plane? 3. \textbf{Zeros}: Are there any points or lines where the field is zero (i.e., the arrows vanish)?

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The question asks to identify the correct graphical representation of the vector field \(\mathbf{A}(x, y) = y\hat{i}\). A vector field assigns a vector (with magnitude and direction) to every point in space. We need to analyze the properties of the given vector field and match them with the figures. 
Step 2: Key Formula or Approach:  
We analyze the vector field \(\mathbf{A} = y\hat{i}\) based on its direction and magnitude at different points in the \(xy\)-plane. 
 

• Direction: The direction of the vector is determined by the sign of its components.
• Magnitude: The magnitude of the vector is \(|\mathbf{A}| = \sqrt{(y)^2 + (0)^2 + (0)^2} = |y|\).

Step 3: Detailed Explanation: 
Let's analyze the vector field \(\mathbf{A} = y\hat{i}\) in detail: 
1. Direction of Vectors:

• The vector field has only an \(\hat{i}\) (x-component). This means all vectors must be horizontal, pointing either to the right (\(+\hat{i}\)) or to the left (\(-\hat{i}\)). All four figures show horizontal vectors.
• The x-component is \(y\).
• When \(y > 0\) (i.e., above the x-axis), the component is positive, so the vectors should point to the right (\(+\hat{i}\)).
• When \(y < 0\) (i.e., below the x-axis), the component is negative, so the vectors should point to the left (\(-\hat{i}\)).
• When \(y = 0\) (i.e., on the x-axis), the component is zero, so the vector is a zero vector (a point).

2. Magnitude of Vectors:

• The magnitude is \(|\mathbf{A}| = |y|\). This means the length of the vector arrows should be proportional to the distance from the x-axis.
• As we move away from the x-axis in either the positive or negative y-direction, the magnitude \(|y|\) increases, so the arrows should become longer.

Now let's examine the options:

• Figure (A):
  • For \(y > 0\), arrows point right. (Correct)
  • For \(y < 0\), arrows point left. (Correct)
  • As \(|y|\) increases, the arrows get longer. (Correct)
  • This figure correctly represents the vector field \(\mathbf{A} = y\hat{i}\).
• Figure (B):
  • For \(y < 0\), arrows point right. (Incorrect, they should point left).
• Figure (C):
  • For \(y > 0\), arrows point left. (Incorrect, they should point right).
• Figure (D):
  • The directions are correct (right for \(y > 0\), left for \(y < 0\)).
  • However, as \(|y|\) increases, the arrows get shorter. This is incorrect, as the magnitude should increase.

Step 4: Final Answer: 
Based on the analysis of direction and magnitude, only Figure (A) provides a correct representation of the vector field \(\mathbf{A} = y\hat{i}\).