Q.Write two different vectors having same magnitude.

Concept: Vector Magnitude Properties — two vectors can have the same length even if their directions are completely different.

Reasoning:

1. The magnitude of a vector $\vec{a} = a_x \hat{i} + a_y \hat{j}$ is $|\vec{a}| = \sqrt{a_x^2 + a_y^2}$.
2. To get the same magnitude, choose any two distinct pairs $(a_x, a_y)$ that give the same sum of squares.
3. For example, take $\vec{p} = 3\hat{i} + 4\hat{j}$ and $\vec{q} = 4\hat{i} + 3\hat{j}$.

Both have magnitude $\sqrt{3^2 + 4^2} = \sqrt{4^2 + 3^2} = 5$, but they point in different directions.

The magnitude (or length) of a vector $\vec{v} = (x, y, z)$ is given by $|\vec{v}| = \sqrt{x^2 + y^2 + z^2}$. This formula depends only on the squares of the components. That means if you change the sign of any component, or rearrange the components, the sum of squares stays the same — as long as the set of absolute values is unchanged.

So to get two different vectors with the same magnitude, you can:

1. Flip signs: Take $\vec{a} = (1, 2, 3)$. Its magnitude is $\sqrt{1^2 + 2^2 + 3^2} = \sqrt{14}$. Now take $\vec{b} = (-1, 2, 3)$. Its magnitude is $\sqrt{(-1)^2 + 2^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14}$. They are different vectors (point in opposite directions along the x-axis) but have the same length.

2. Permute components: Another pair: $\vec{c} = (1, 2, 3)$ and $\vec{d} = (3, 1, 2)$. Both have magnitude $\sqrt{14}$ because the sum of squares is $1+4+9$ in both cases.

Method: Constructing two different vectors with the same magnitude

Use this "give an example" task by exploiting that magnitude depends only on the squares of components.

Steps

Step 1: Recall what fixes the magnitude

Only the sum of squares matters — not the order of the components or their signs.

Step 2: Keep the sum of squares fixed while changing the vector

Two easy ways to produce a genuinely different vector with the same length:

• Permute the components, e.g. $3\hat{i}+4\hat{j}$ and $4\hat{i}+3\hat{j}$ (both give $\sqrt{25}=5$).
• Flip a sign, e.g. $(1,2,3)$ and $(-1,2,3)$ (both give $\sqrt{14}$).

Step 3: Verify and confirm they are truly different

Compute both magnitudes to check they match, and confirm the two vectors are not identical (and ideally not just negatives of each other, so the "different" is unmistakable).