Which one of the following statements is false regarding the vectors ?

Options

A. The magnitude of a vector is always a scalar.
B. Each component of a vector is always a scalar.
C. Two vectors having different magnitudes cannot have their resultant zero.
D. Vectors obey triangle law of addition.

Explanation

To determine which statement is false regarding vectors, we will analyze each option provided in the question step by step.

### Step 1: Analyze the first statement
**Statement:** The magnitude of a vector is always a scalar.
**Analysis:** The magnitude of any vector is indeed a scalar quantity. For example, if we have a vector represented as \( \mathbf{A} = 1\hat{i} + 2\hat{j} - 1\hat{k} \), the magnitude is calculated as:
\[ |\mathbf{A}| = \sqrt{(1)^2 + (2)^2 + (-1)^2} = \sqrt{1 + 4 + 1} = \sqrt{6} \]
Since \( \sqrt{6} \) is a scalar, this statement is true.

### Step 2: Analyze the second statement
**Statement:** Each component of a vector is always a scalar.
**Analysis:** In the vector \( \mathbf{A} = 1\hat{i} + 2\hat{j} - 1\hat{k} \), the components are \( 1 \), \( 2 \), and \( -1 \). All these components are scalar quantities. Thus, this statement is also true.

### Step 3: Analyze the third statement
**Statement:** Vectors obey the triangle law of vector addition.
**Analysis:** According to the triangle law, if two vectors are represented as two sides of a triangle, the resultant vector can be represented as the third side. This is a fundamental property of vector addition. Therefore, this statement is true.

### Step 4: Analyze the fourth statement
**Statement:** Two vectors having different magnitudes cannot have their resultant zero.
**Analysis:** This statement is false. Two vectors can have different magnitudes and still result in a zero vector if they are equal in magnitude but opposite in direction. For example, if we have two vectors \( \mathbf{A} = 3\hat{i} \) and \( \mathbf{B} = -3\hat{i} \), their resultant is:
\[ \mathbf{A} + \mathbf{B} = 3\hat{i} - 3\hat{i} = 0 \]
Thus, this statement is indeed false.

### Conclusion
The false statement regarding vectors is: **"Two vectors having different magnitudes cannot have their resultant zero."**

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