Given points $A(-1, -3)$ and $B(0, -1)$, and vector $\vec{a} = (2, y)$. Also, $\vec{AB} // \vec{a}$. Find the value of $y$.

1. Problem Description

Given points $A(-1, -3)$ and $B(0, -1)$, and vector $\vec{a} = (2, y)$. Also, $\vec{AB} // \vec{a}$. Find the value of $y$.

2. Solution Steps

First, we need to find the vector $\vec{AB}$.

$\vec{AB} = B - A = (0 - (-1), -1 - (-3)) = (1, 2)$

Since $\vec{AB} // \vec{a}$, there exists a scalar $k$ such that $\vec{AB} = k\vec{a}$. Thus,

$(1, 2) = k(2, y) = (2k, ky)$

From the first component, we have $1 = 2k$, so $k = \frac{1}{2}$.

From the second component, we have $2 = ky$. Substituting $k = \frac{1}{2}$, we get $2 = \frac{1}{2}y$.

Solving for $y$, we have $y = 2 \cdot 2 = 4$.

3. Final Answer

The value of $y$ is

4. Answer: B. 4