Obtain the condition for the two vectors `vec(A) = x_(1) hat(i) + y_(1)hat(j) + z_(1) hat(k) and vec(B) = x_(2) hat(i) = x_(2) hat(i) + y_(2) hat(j) + z_(2) hat(k)` to be parallel ?

Explanation

To determine the condition under which the two vectors \(\vec{A} = x_1 \hat{i} + y_1 \hat{j} + z_1 \hat{k}\) and \(\vec{B} = x_2 \hat{i} + y_2 \hat{j} + z_2 \hat{k}\) are parallel, we can follow these steps:

### Step 1: Understand the condition for parallel vectors
Two vectors are parallel if their cross product is zero. Therefore, we need to find the condition under which \(\vec{A} \times \vec{B} = \vec{0}\).

### Step 2: Calculate the cross product
The cross product \(\vec{A} \times \vec{B}\) can be computed using the determinant of a matrix formed by the unit vectors and the components of the vectors:

\[ \vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \end{vmatrix} \]

### Step 3: Expand the determinant
To calculate the determinant, we can use the rule of Sarrus or cofactor expansion:

\[ \vec{A} \times \vec{B} = \hat{i} \begin{vmatrix} y_1 & z_1 \\ y_2 & z_2 \end{vmatrix} - \hat{j} \begin{vmatrix} x_1 & z_1 \\ x_2 & z_2 \end{vmatrix} + \hat{k} \begin{vmatrix} x_1 & y_1 \\ x_2 & y_2 \end{vmatrix} \]

Calculating each of these determinants gives us:

\[ \vec{A} \times \vec{B} = \hat{i} (y_1 z_2 - z_1 y_2) - \hat{j} (x_1 z_2 - z_1 x_2) + \hat{k} (x_1 y_2 - y_1 x_2) \]

### Step 4: Set the cross product equal to zero
For the vectors to be parallel, we set the components of the cross product to zero:

1. \(y_1 z_2 - z_1 y_2 = 0\)
2. \(x_1 z_2 - z_1 x_2 = 0\)
3. \(x_1 y_2 - y_1 x_2 = 0\)

### Step 5: Solve the equations
From these equations, we can derive the ratios:

1. From \(y_1 z_2 - z_1 y_2 = 0\), we get:
\[ \frac{y_1}{y_2} = \frac{z_1}{z_2} \]

2. From \(x_1 z_2 - z_1 x_2 = 0\), we get:
\[ \frac{x_1}{x_2} = \frac{z_1}{z_2} \]

3. From \(x_1 y_2 - y_1 x_2 = 0\), we get:
\[ \frac{x_1}{x_2} = \frac{y_1}{y_2} \]

### Step 6: Combine the ratios
Combining these ratios, we find that:

\[ \frac{x_1}{x_2} = \frac{y_1}{y_2} = \frac{z_1}{z_2} \]

### Conclusion
Thus, the condition for the vectors \(\vec{A}\) and \(\vec{B}\) to be parallel is:

\[ \frac{x_1}{x_2} = \frac{y_1}{y_2} = \frac{z_1}{z_2} \]

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