Prove that if `vec(u)=u_(1)hat(1)hat(i)+u_(2)hat(j)` and `vec(v)=v_(1)hat(i)+v_(2)hat(j)` are non - zero vectors, then they are parallel if and only if `u_(1)v_(2)-u_(2)v_(1)=0`.

Answer

To prove that the vectors \(\vec{u} = u_1 \hat{i} + u_2 \hat{j}\) and \(\vec{v} = v_1 \hat{i} + v_2 \hat{j}\) are parallel if and only if \(u_1 v_2 - u_2 v_1 = 0\), we will follow these steps:

### Step 1: Understanding the Condition for Parallel Vectors
Two vectors \(\vec{u}\) and \(\vec{v}\) are parallel if one is a scalar multiple of the other. This means there exists a scalar \(\lambda\) such that:
\[ \vec{u} = \lambda \vec{v} \]

### Step 2: Expressing the Vectors
From the given vectors:
\[ \vec{u} = u_1 \hat{i} + u_2 \hat{j} \]
\[ \vec{v} = v_1 \hat{i} + v_2 \hat{j} \]

### Step 3: Setting Up the Equation
If \(\vec{u}\) is a scalar multiple of \(\vec{v}\), we can write:
\[ u_1 \hat{i} + u_2 \hat{j} = \lambda (v_1 \hat{i} + v_2 \hat{j}) \]

### Step 4: Equating Components
By equating the components of \(\hat{i}\) and \(\hat{j}\), we get:
1. \(u_1 = \lambda v_1\)
2. \(u_2 = \lambda v_2\)

### Step 5: Finding the Ratio
From the equations above, we can express \(\lambda\) in terms of the components:
\[ \lambda = \frac{u_1}{v_1} = \frac{u_2}{v_2} \]

### Step 6: Cross Multiplying
Cross multiplying the ratios gives us:
\[ u_1 v_2 = u_2 v_1 \]

### Step 7: Rearranging the Equation
Rearranging this equation leads us to:
\[ u_1 v_2 - u_2 v_1 = 0 \]

### Step 8: Conclusion
Thus, we have shown that the vectors \(\vec{u}\) and \(\vec{v}\) are parallel if and only if:
\[ u_1 v_2 - u_2 v_1 = 0 \]