Let's break down where the point $(0, 1, -1)$ comes from and how it's used in the solution for finding the equation of the plane.

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Step 1: Understanding the Line Equation

The given line is described by the symmetric equations: $x = \frac{y - 1}{2} = z + 1$

This implies the parametric equations of the line:

$x = t, \quad y = 2t + 1, \quad z = t - 1$

Here, $t$ is a parameter.

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Step 2: Identify a Point on the Line

We can find a specific point on the line by choosing a convenient value for $t$. Let’s select $t = 0$:

$x = 0, \quad y = 2(0) + 1 = 1, \quad z = 0 - 1 = -1$

Therefore, the point $(0, 1, -1)$ lies on the line.

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Step 3: Purpose of the Point in the Solution

The point $(0, 1, -1)$ is used along with the given point $(1, 4, 3)$ to define a vector parallel to the plane. This vector is:

$\mathbf{v_2} = (1 - 0, 4 - 1, 3 - (-1)) = \langle 1, 3, 4 \rangle$

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Step 4: Cross Product to Find the Normal Vector

The plane must contain both vectors:

• $\mathbf{v_1} = \langle 1, 2, 1 \rangle$ (the direction vector of the line)
• $\mathbf{v_2} = \langle 1, 3, 4 \rangle$ (the vector between two points)

The cross product of these two vectors gives the normal vector to the plane:

\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 2 & 1 \\ 1 & 3 & 4 \end{vmatrix}$$ $$= \mathbf{i} (8 - 3) - \mathbf{j} (4 - 1) + \mathbf{k} (3 - 2) = 5\mathbf{i} - 3\mathbf{j} + \mathbf{k}$$ --- ### **Step 5: Equation of the Plane** Now that we have the normal vector $$\mathbf{n} = \langle 5, -3, 1 \rangle$$ and a point on the plane $$(1, 4, 3)$$, the scalar equation for the plane is: $$5(x - 1) - 3(y - 4) + 1(z - 3) = 0$$ Simplifying: $$5x - 3y + z + 4 = 0$$ --- ### **Summary** The point $$(0, 1, -1)$$ was obtained by setting the parameter $$t = 0$$ in the parametric equations of the given line. It served as a convenient point to define the vector $$\mathbf{v_2}$$, which is crucial for finding the normal vector of the plane through the cross product. Would you like a deeper dive into the cross product calculation or any specific step?