Question

Let \(\vec{a} = \hat{i} + 2\hat{j}\) and \(\vec{b} = 2\hat{i} + \hat{j}\). Which of the following statements holds true?

• A. \(|\vec{a}| = |\vec{b}|\), \(\vec{a}\) is not equal to \(\vec{b}\)
• B. \(|\vec{a}| \neq |\vec{b}|\), \(\vec{a}\) is equal to \(\vec{b}\)
• C. \(|\vec{a}| = |\vec{b}|\), \(\vec{a}\) is equal to \(\vec{b}\)
• D. \(|\vec{a}| \neq |\vec{b}|\), \(\vec{a}\) is not equal to \(\vec{b}\)

Hint:

Equal magnitude does NOT imply equal vectors. Direction depends on orientation (\(\tan\theta_a = 2/1 = 2\) vs \(\tan\theta_b = 1/2\)).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Two vectors are equal if and only if their corresponding orthogonal components are identical.
Two vectors have equal magnitude if the square root of the sum of the squares of their components are equal.
Key Formula or Approach:
\[ |\vec{v}| = \sqrt{v_x^2 + v_y^2} \]
\[ \vec{a} = \vec{b} \iff a_x = b_x \text{ and } a_y = b_y \]

Step 2: Detailed Explanation:
Calculating the magnitude of vector \(\vec{a}\):
\[ |\vec{a}| = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \]
Calculating the magnitude of vector \(\vec{b}\):
\[ |\vec{b}| = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5} \]
Since \(|\vec{a}| = \sqrt{5}\) and \(|\vec{b}| = \sqrt{5}\), their magnitudes are equal:
\[ |\vec{a}| = |\vec{b}| \]
Comparing the vector components:
The \(x\)-component of \(\vec{a}\) is \(1\), while that of \(\vec{b}\) is \(2\) (\(1 \neq 2\)).
Therefore, \(\vec{a} \neq \vec{b}\).

Step 3: Final Answer:
Hence, \(|\vec{a}| = |\vec{b}|\) and \(\vec{a}\) is not equal to \(\vec{b}\), which corresponds to option (A).