Find a parametric representation for the plane that passes through the point $(1, -2, 1)$ and contains the vectors $\mathbf{i} + \mathbf{j} - \mathbf{k}$ and $\mathbf{i} - \mathbf{j} + \mathbf{k}$ . (Enter your answer as a comma-separated list of equations. Let $x$ , $y$ , and $z$ be in terms of $u$ and/or $v$ .)

Question :

Find a parametric representation for the plane that passes through the point $(1, -2, 1)$ and contains the vectors $\mathbf{i} + \mathbf{j} - \mathbf{k}$ and $\mathbf{i} - \mathbf{j} + \mathbf{k}$ . (enter your answer as a comma-separated list of equations. let $x$ , $y$ , and $z$ be in terms of $u$ and/or $v$ .)

Find a parametric representation for the plane that passes through the point $(1 | Doubtlet.com

Solution:

Neetesh Kumar | November 14, 2024

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This is the solution to Math 1D
Assignment: 16.6 Question Number 12
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Step-by-step solution:

To find the parametric representation of a plane passing through a point and containing two direction vectors, we use the following approach.

Step 1: Identify the Point and Direction Vectors

The given point through which the plane passes is: $(1, -2, 1)$

The two given direction vectors are: $\mathbf{d_1} = \langle 1, 1, -1 \rangle$ $\mathbf{d_2} = \langle 1, -1, 1 \rangle$

Step 2: Write the Parametric Equation of the Plane

A parametric equation for a plane passing through a point $\mathbf{P_0} = (1, -2, 1)$ with direction vectors $\mathbf{d_1}$ and $\mathbf{d_2}$ can be written as: $\mathbf{r}(u, v) = \mathbf{P_0} + u \mathbf{d_1} + v \mathbf{d_2}$

Substituting $\mathbf{P_0}$ , $\mathbf{d_1}$ , and $\mathbf{d_2}$ : $\mathbf{r}(u, v) = \langle 1, -2, 1 \rangle + u \langle 1, 1, -1 \rangle + v \langle 1, -1, 1 \rangle$

Step 3: Expand to Find $x$ , $y$ , and $z$ Components

Now, let’s expand each component individually:

For the $x$ -component: $x = 1 + u \cdot 1 + v \cdot 1$ $x = 1 + u + v$
For the $y$ -component: $y = -2 + u \cdot 1 + v \cdot (-1)$ $y = -2 + u - v$
For the $z$ -component: $z = 1 + u \cdot (-1) + v \cdot 1$ $z = 1 - u + v$

Final Parametric Equations

The parametric representation for the plane is: $x = 1 + u + v, \quad y = -2 + u - v, \quad z = 1 - u + v$

Answer:

The parametric equations are: $x = 1 + u + v, \quad y = -2 + u - v, \quad z = 1 - u + v$

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