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

Name: Problem 269, Chapter 2: Calculus Volume 3
Uploaded: 2019-10-06T05:46:26-08:00
Duration: 45 s
Channel: Jie Min
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(1,2,3), 𝐧=⟨1,2,3⟩

step 1 Here's the formula, we know that the equation will be given by 1, 2, 3, which is normal vector t

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 $\mathbf{n} .$ $P(1,2,3), \quad \mathbf{n}=\langle 1,2,3\rangle$

   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 $\mathbf{n} .$ $P(1,2,3), \quad \mathbf{n}=\langle 1,2,3\rangle$

Calculus Volume 3

Gilbert Strang 1st Edition

Chapter 2, Problem 269

Instant Answer

Solved by Expert Jie Min

10/06/2019

Step 1

In this case, $\mathbf{n}=\langle 1,2,3\rangle$ and $\mathbf{r_0}=\langle 1,2,3\rangle$. 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 $\mathbf{n} .$ $P(1,2,3), \quad \mathbf{n}=\langle 1,2,3\rangle$

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Key Concepts

Scalar (or General) Equation of a Plane

This is the expanded version of the point-normal form, where the dot product is fully computed to obtain an expression in the variables x, y, and z. The resulting equation is typically written in the form Ax + By + Cz + D = 0, where A, B, and C are the components of the normal vector and D is obtained by incorporating the coordinates of the known point. This format is useful for linear algebra and analytic geometry applications.

Point-Normal Form of a Plane

This concept refers to the equation form of a plane derived from a given point on the plane and a normal vector. The equation is based on the idea that any vector lying on the plane, when subtracted from the given point, will be orthogonal to the normal vector. This orthogonality condition is encapsulated by the dot product equaling zero, which forms the foundation of the point-normal form equation.

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