4. The following equation describes a straight line: $$ \langle x, y, z\rangle=\langle-1,0,-2\rangle+t\langle 1,2,2\rangle $$ Find the coordinates of the point where this line intersects the $$ y $$-axis.

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To find the point where the given line intersects the $$ y $$-axis, let's analyze the parametric equations of the line:

$$
\langle x, y, z \rangle = \langle -1, 0, -2 \rangle + t \langle 1, 2, 2 \rangle
$$

This can be broken down into the following component equations:

$$
\begin{cases}
x = -1 + t \
y = 0 + 2t \
z = -2 + 2t
\end{cases}
$$

**Intersection with the $$ y $$-axis:**

On the $$ y $$-axis in 3D space, the coordinates satisfy $$ x = 0 $$ and $$ z = 0 $$. Therefore, we set the corresponding component equations to zero and solve for $$ t $$:

1. **For $$ x = 0 $$:**
   $$
   -1 + t = 0 \implies t = 1
   $$

2. **For $$ z = 0 $$:**
   $$
   -2 + 2t = 0 \implies 2t = 2 \implies t = 1
   $$

Both conditions yield $$ t = 1 $$. Now, substitute $$ t = 1 $$ back into the $$ y $$-component equation to find the $$ y $$-coordinate:

$$
y = 0 + 2(1) = 2
$$

**Conclusion:**
The line intersects the $$ y $$-axis at the point $$(0, 2, 0)$$.

**Answer:**  
After simplifying, the intersection point is (0 , 2, 0).
The line intersects the $$ y $$-axis at the point $$(0, 2, 0)$$.

4. The following equation describes a straight line: \[ \langle x, y, z\rangle=\langle-1,0,-2\rangle+t\langle 1,2,2\rangle \] Find the coordinates of the point where this line intersects the \( y \)-axis.

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