Find the point at which the line $$ \vec{r}(t)=\langle 1,2,4\rangle+t\langle-3,7,-1\rangle $$ intersects the $$ x y $$ plane.

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To find the point where the line $$ \vec{r}(t) = \langle 1, 2, 4 \rangle + t\langle -3, 7, -1 \rangle $$ intersects the $$ xy $$-plane, follow these steps:

1. **Identify the $$ z $$-component:**
   The $$ xy $$-plane is defined by $$ z = 0 $$. Therefore, we need to find the parameter $$ t $$ for which the $$ z $$-component of $$ \vec{r}(t) $$ is zero.

2. **Set the $$ z $$-component to zero:**
   $$
   z(t) = 4 + t(-1) = 0
   $$
   $$
   4 - t = 0 \implies t = 4
   $$

3. **Find the $$ x $$ and $$ y $$ coordinates at $$ t = 4 $$:**
   $$
   x(t) = 1 + t(-3) = 1 - 3(4) = 1 - 12 = -11
   $$
   $$
   y(t) = 2 + t(7) = 2 + 7(4) = 2 + 28 = 30
   $$

4. **Write the intersection point:**
   $$
   \text{Intersection Point} = (-11, 30, 0)
   $$

**Final Answer:**
The line intersects the xy-plane at the point ( –11, 30, 0 ).
The line intersects the $$ xy $$-plane at the point (–11, 30, 0).

Find the point at which the line intersects the plane.

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