For the following exercises, point P and vector 𝐧 are given. a. Find the scalar equation of the

Name: Problem 268, Chapter 2: Calculus Volume 3
Uploaded: 2019-08-05T20:31:10-08:00
Duration: 48 s
Channel: Amy Jiang
Description: For the following exercises, point P and vector 𝐧 are given. a. Find the scalar equation of the plane that passes through P and has normal vector 𝐧 . b. Find the general form of the equation of the plane that passes through P and has normal vector 𝐧 . P(3,2,2), 𝐧=2 𝐢+3 𝐣-𝐤

step 1 Okay, so we're giving that p is equal to 3, 2, 2, and is equal to 2 i plus 3 j minus k. We have

Question

For the following exercises, point $P$ and vector $\mathbf{n}$ are given. a. Find the scalar equation of the plane that passes through $P$ and has normal vector $\mathbf{n} .$ b. Find the general form of the equation of the plane that passes through $P$ and has normal vector $\mathbf{n} .$ $P(3,2,2), \quad \mathbf{n}=2 \mathbf{i}+3 \mathbf{j}-\mathbf{k}$

   For the following exercises, point $P$ and vector $\mathbf{n}$ are given. 
a. Find the scalar equation of the plane that passes through $P$ and has normal vector $\mathbf{n} .$ 
b. Find the general form of the equation of the plane that passes through $P$ and has normal vector $\mathbf{n} .$ $P(3,2,2), \quad \mathbf{n}=2 \mathbf{i}+3 \mathbf{j}-\mathbf{k}$

Calculus Volume 3

Gilbert Strang 1st Edition

Chapter 2, Problem 268

Instant Answer

Solved by Expert Amy Jiang

08/05/2019

Step 1

In this case, $\mathbf{n}=2\mathbf{i}+3\mathbf{j}-\mathbf{k}$ and $\mathbf{r_0}=3\mathbf{i}+2\mathbf{j}+2\mathbf{k}$. Show more…

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For the following exercises, point $P$ and vector $\mathbf{n}$ are given. a. Find the scalar equation of the plane that passes through $P$ and has normal vector $\mathbf{n} .$ b. Find the general form of the equation of the plane that passes through $P$ and has normal vector $\mathbf{n} .$ $P(3,2,2), \quad \mathbf{n}=2 \mathbf{i}+3 \mathbf{j}-\mathbf{k}$

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Key Concepts

Plane Equation

A plane in three-dimensional space can be described by a linear equation that every point on the plane satisfies. This equation characterizes a flat, two-dimensional subset of space and is fundamental in analytic geometry.

Normal Vector

A normal vector is a vector that is perpendicular to every vector lying within a plane. It determines the orientation of the plane and is crucial in deriving equations that represent the plane, as it provides the directional information needed for the point-normal form.

Point-Normal Form

The point-normal form of the equation of a plane uses a known point through which the plane passes and a normal vector perpendicular to the plane. This form directly leads to the scalar and general forms of the plane's equation by ensuring that the vector drawn from the given point to any other point in the plane remains orthogonal to the normal vector.

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