Question 5 of 68beginner💡 UnderstandShort Answer1 mark

Write two different vectors having same magnitude.

Correct Answer

Two different vectors having the same magnitude are a⃗=i^+2j^\vec{a} = \hat{i} + 2\hat{j} and b⃗=2i^+j^\vec{b} = 2\hat{i} + \hat{j}.

Both vectors have magnitude ∣a⃗∣=12+22=5|\vec{a}| = \sqrt{1^2 + 2^2} = \sqrt{5} and ∣b⃗∣=22+12=5|\vec{b}| = \sqrt{2^2 + 1^2} = \sqrt{5}. However, the two vectors are not equal since their corresponding components are distinct.

Exercise: EXERCISE 10.2 | Q: 2 | (Page 17)
For More Understanding

Explanation

This question is directly addressed in Example 5 of the textbook. The example demonstrates that two vectors can have the same magnitude but still be different vectors if their corresponding components are different. The magnitude of a vector depends on the sum of squares of its components, so different combinations can yield the same result.

Solution Steps

  1. Step 1: Choose two vectors with different components, e.g., a⃗=i^+2j^\vec{a} = \hat{i} + 2\hat{j} and b⃗=2i^+j^\vec{b} = 2\hat{i} + \hat{j}

  2. Step 2: Calculate magnitude of a⃗\vec{a}: ∣a⃗∣=12+22=5|\vec{a}| = \sqrt{1^2 + 2^2} = \sqrt{5}

  3. Step 3: Calculate magnitude of b⃗\vec{b}: ∣b⃗∣=22+12=5|\vec{b}| = \sqrt{2^2 + 1^2} = \sqrt{5}

  4. Step 4: Verify that ∣a⃗∣=∣b⃗∣|\vec{a}| = |\vec{b}| but the vectors are different since their components are not equal.

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