A line passes through the distinct vectors $\mathbf{a}$ and $\mathbf{b}.$  Which of the following are valid parameterizations of the line?

(A) $\mathbf{a} + t (\mathbf{a} - \mathbf{b})$

(B) $\frac{3}{4} \mathbf{a} + t \cdot \frac{\mathbf{b}}{4}$

(C) $\frac{\mathbf{a} + \mathbf{b}}{2} + t \cdot \frac{\mathbf{b} - \mathbf{a}}{2}$

(D) $\mathbf{a} + 2 \mathbf{b} + t (2 \mathbf{b} - 3 \mathbf{a})$

(E) $-\mathbf{b} + t (\mathbf{a} + \mathbf{b})$

(F) $\mathbf{b} + \frac{t}{3} (2 \mathbf{a} - 2 \mathbf{b})$

Enter the letters of the correct options, separated by commas.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
A line passes through the distinct vectors $\mathbf{a}$ and $\mathbf{b}.$  Which of the following are valid parameterizations of the line?

(A) $\mathbf{a} + t (\mathbf{a} - \mathbf{b})$

(B) $\frac{3}{4} \mathbf{a} + t \cdot \frac{\mathbf{b}}{4}$

(C) $\frac{\mathbf{a} + \mathbf{b}}{2} + t \cdot \frac{\mathbf{b} - \mathbf{a}}{2}$

(D) $\mathbf{a} + 2 \mathbf{b} + t (2 \mathbf{b} - 3 \mathbf{a})$

(E) $-\mathbf{b} + t (\mathbf{a} + \mathbf{b})$

(F) $\mathbf{b} + \frac{t}{3} (2 \mathbf{a} - 2 \mathbf{b})$

Enter the letters of the correct options, separated by commas.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\text{A,C,F}
A line passes through the distinct vectors $\mathbf{a}$ and $\mathbf{b}.$  Which of the following are valid parameterizations of the line?

(A) $\mathbf{a} + t (\mathbf{a} - \mathbf{b})$

(B) $\frac{3}{4} \mathbf{a} + t \cdot \frac{\mathbf{b}}{4}$

(C) $\frac{\mathbf{a} + \mathbf{b}}{2} + t \cdot \frac{\mathbf{b} - \mathbf{a}}{2}$

(D) $\mathbf{a} + 2 \mathbf{b} + t (2 \mathbf{b} - 3 \mathbf{a})$

(E) $-\mathbf{b} + t (\mathbf{a} + \mathbf{b})$

(F) $\mathbf{b} + \frac{t}{3} (2 \mathbf{a} - 2 \mathbf{b})$

Enter the letters of the correct options, separated by commas.
train