A particle is moving with initial velocity `bar(u)=hati-hatj+hatk`. What should be its acceleration so that it can remain moving in the same straight line?

Options

A. `vec(a)=2hati-2hatj+2hatk`
B. `vec(a)=-2hati+2hatj+2hatk`
C. `vec(a)=3hati+3hatj+3hatk`
D. `vec(a)=1hati-1hatj`

Explanation

To solve the problem, we need to determine the acceleration required for a particle to continue moving in the same straight line as its initial velocity. The initial velocity of the particle is given as:

\[ \vec{u} = \hat{i} - \hat{j} + \hat{k} \]

### Step 1: Understand the condition for straight-line motion
For a particle to continue moving in the same straight line, its acceleration must be in the same direction as its velocity. This means that the acceleration vector must be a scalar multiple of the velocity vector.

### Step 2: Identify the direction of the initial velocity
The initial velocity vector \(\vec{u}\) can be expressed in component form as:
\[ \vec{u} = (1, -1, 1) \]
This indicates that the particle is moving in the direction of the vector \((1, -1, 1)\).

### Step 3: Determine the acceleration vector
Let the acceleration vector be represented as:
\[ \vec{a} = a_x \hat{i} + a_y \hat{j} + a_z \hat{k} \]
For the particle to maintain its straight-line motion, the acceleration vector \(\vec{a}\) must be in the same direction as \(\vec{u}\). Therefore, we can express \(\vec{a}\) as a scalar multiple of \(\vec{u}\):
\[ \vec{a} = k \cdot \vec{u} \]
where \(k\) is a scalar.

### Step 4: Write the acceleration in terms of the initial velocity
Substituting \(\vec{u}\) into the equation gives:
\[ \vec{a} = k(\hat{i} - \hat{j} + \hat{k}) = k\hat{i} - k\hat{j} + k\hat{k} \]
This shows that the acceleration vector has the same direction as the initial velocity vector.

### Step 5: Conclusion
Thus, the acceleration can be expressed as:
\[ \vec{a} = k(\hat{i} - \hat{j} + \hat{k}) \]
for any scalar \(k\). This indicates that the acceleration must be in the direction of \(\hat{i} - \hat{j} + \hat{k}\) for the particle to continue moving in the same straight line.