Introduction to Relative Velocity
Relative velocity is a fundamental concept in kinematics that describes the velocity of one object as observed from the frame of reference of another object. In 3D space, we represent these velocities as vectors using the unit vectors $\hat{i}$, $\hat{j}$, and $\hat{k}$, which correspond to the $x$, $y$, and $z$ axes respectively.
The Concept
If we have two bodies, A and B, with velocities $\vec{V}_A$ and $\vec{V}_B$, the velocity of B relative to A (denoted as $\vec{V}_{BA}$) is defined as the difference between the velocity of B and the velocity of A:
$$\vec{V}_{BA} = \vec{V}_B - \vec{V}_A$$
Step-by-Step Solution
Given the velocities:
- $\vec{V}_A = \hat{i} + 2\hat{j} - 3\hat{k}$
- $\vec{V}_B = 3\hat{i} + 2\hat{j} - \hat{k}$
We need to find $\vec{V}_{BA} = \vec{V}_B - \vec{V}_A$.
Step 1: Write down the subtraction equation. $$\vec{V}_{BA} = (3\hat{i} + 2\hat{j} - \hat{k}) - (\hat{i} + 2\hat{j} - 3\hat{k})$$
Step 2: Group the unit vector components. $$\vec{V}_{BA} = (3 - 1)\hat{i} + (2 - 2)\hat{j} + (-1 - (-3))\hat{k}$$
Step 3: Perform the arithmetic.
- $x$-component: $3 - 1 = 2$
- $y$-component: $2 - 2 = 0$
- $z$-component: $-1 + 3 = 2$
Final Result: $$\vec{V}_{BA} = 2\hat{i} + 0\hat{j} + 2\hat{k} = 2\hat{i} + 2\hat{k}$$
Intuition
Think of relative velocity as how much faster or in what different direction B is moving compared to A. By subtracting $\vec{V}_A$, you are effectively placing yourself in A's frame of reference, making A appear stationary (at zero velocity). In this specific example, the $y$-components are identical ($2\hat{j}$), so relative to A, B has no movement along the $y$-axis.