write the equation of a plane passing through p with direction vectos u and v in vector and parametric forms



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.

write the equation of a plane passing through p with direction vectors u and v in vector and parametric forms

Explore how to find the vector and parametric equations of a plane passing through a point P with two given direction vectors. This problem demonstrates the key formulas for representing planes in 3D space, including the vector form and parametric form of a plane equation.

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