To find the equation of a plane passing through a point $P(x_0, y_0, z_0)$ and containing two direction vectors $\mathbf{u} = \langle u_1, u_2, u_3 \rangle$ and $\mathbf{v} = \langle v_1, v_2, v_3 \rangle$, we can represent the plane in vector form and parametric form as follows:

1. Vector Form:

The vector form of a plane can be written as: $\mathbf{r}(s, t) = \mathbf{r}_0 + s\mathbf{u} + t\mathbf{v}$ Where:

• $\mathbf{r}(s, t)$ is a general point on the plane.
• $\mathbf{r}_0 = \langle x_0, y_0, z_0 \rangle$ is the position vector of point $P$.
• $\mathbf{u} = \langle u_1, u_2, u_3 \rangle$ and $\mathbf{v} = \langle v_1, v_2, v_3 \rangle$ are the direction vectors.
• $s$ and $t$ are scalar parameters.

So, the vector form of the plane is: $\mathbf{r}(s, t) = \langle x_0, y_0, z_0 \rangle + s \langle u_1, u_2, u_3 \rangle + t \langle v_1, v_2, v_3 \rangle$

2. Parametric Form:

Expanding the vector form into coordinates, we get the parametric equations for the plane: $x = x_0 + s u_1 + t v_1$ $y = y_0 + s u_2 + t v_2$ $z = z_0 + s u_3 + t v_3$

Here, $s$ and $t$ are free parameters that can take any real values, determining the coordinates of any point on the plane.

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Would you like more details or have any questions about these forms?

Related Questions:

1. How can I find the normal vector to the plane using the direction vectors $\mathbf{u}$ and $\mathbf{v}$?
2. What is the general form of a plane equation if we have the normal vector?
3. How do I calculate the angle between two planes?
4. Can you explain how to find the intersection of a line with the plane?
5. How do I find the distance between a point and the plane?

Tip:

The cross product of the two direction vectors $\mathbf{u}$ and $\mathbf{v}$ gives the normal vector to the plane.