Vector `vec(A)=hat(i)+hat(j)-2hat(k)` and `vec(B)=3hat(i)+3hat(j)-6hat(k)` are `:`

Options

A. Parallel
B. Antiparallel
C. Perpendicular
D. at acute angle with each other

Explanation

To determine the relationship between the vectors \(\vec{A}\) and \(\vec{B}\), we can follow these steps:

### Step 1: Write down the vectors
Given:
\[ \vec{A} = \hat{i} + \hat{j} - 2\hat{k} \]
\[ \vec{B} = 3\hat{i} + 3\hat{j} - 6\hat{k} \]

### Step 2: Factor out the common components in \(\vec{B}\)
We can factor out the common factor from \(\vec{B}\):
\[ \vec{B} = 3(\hat{i} + \hat{j} - 2\hat{k}) \]

### Step 3: Relate \(\vec{B}\) to \(\vec{A}\)
Notice that:
\[ \vec{A} = \hat{i} + \hat{j} - 2\hat{k} \]
Thus, we can express \(\vec{B}\) in terms of \(\vec{A}\):
\[ \vec{B} = 3\vec{A} \]

### Step 4: Determine the relationship
Since \(\vec{B} = 3\vec{A}\), we can conclude that \(\vec{B}\) is a scalar multiple of \(\vec{A}\) with a positive scalar (3). This indicates that the two vectors are parallel.

### Step 5: Conclusion
The vectors \(\vec{A}\) and \(\vec{B}\) are parallel, and the angle between them is \(0^\circ\).