# Vector and Plane Relationship in 3D

• I
• Travis Enigma
In summary, the normal vector of a plane is always perpendicular to all vectors contained in the plane. This can be defined by stating that the dot product between the normal vector and any vector in the plane is always equal to zero. In geometric terms, this means that the normal vector is constrained to be perpendicular to the plane.
Travis Enigma
TL;DR Summary
Vector contained inside the plane.
I have a quick question. If a Vector is contained inside a plane, would the normal of the plane be orthogonal to said vector?

Thank you!

Yes. The normal vector of the plane is perpendicular to all vectors in the plane.

jedishrfu and Travis Enigma
Okay thank you so much!

fresh_42 said:
Yes. The normal vector of the plane is perpendicular to all vectors in the plane.
Does that statement relate to some axiom / definition? Pardon my non-mathematical ignorance but what constrains the normal not to be anywhere in 3D? (I'm thinking Geometry here)

Let ##P:=\mathbb{R}\cdot \vec{p} \oplus \mathbb{R}\cdot \vec{q}## be the plane and ##\vec{n}## its normal vector. Then ##\langle \vec{n},P \rangle =0## by definition of normality. This means that ##\langle \vec{n},\lambda \vec{p}+\mu \vec{q} \rangle=0## for all ##\lambda ,\mu \in \mathbb{R}.## This is especially true for the given vector contained in the plane whose coordinates are a specific pair ##(\lambda ,\mu).##

If you define normality by ##\vec{n} \perp \vec{p}\, \wedge \vec{n}\perp \vec{q} \,,## then we get
$$0=\lambda \cdot 0+\mu\cdot 0=\lambda \langle \vec{n},\vec{p}\rangle +\mu \langle \vec{n},\vec{q}\rangle=\langle \vec{n},\lambda \vec{p}+\mu\vec{q}\rangle$$
and again ##\vec{n}\perp \lambda \vec{p}+\mu\vec{q}## for a given pair of coordinates ##(\lambda ,\mu).##

## 1. What is a vector in 3D space?

A vector in 3D space is a mathematical object that has both magnitude (size) and direction. It is represented by an arrow pointing from its initial point to its terminal point.

## 2. How are vectors and planes related in 3D?

Vectors and planes are related in 3D through the concept of dot product. The dot product of a vector and a normal vector of a plane is equal to the component of the vector in the direction of the normal vector. This allows us to determine if a vector is parallel or perpendicular to a plane, and also to find the angle between them.

## 3. What is the equation of a plane in 3D?

The equation of a plane in 3D is ax + by + cz = d, where a, b, and c are the coefficients of the plane's normal vector and d is a constant term. This equation can also be written in vector form as r · n = d, where r is a position vector and n is the normal vector.

## 4. How do you find the intersection between a vector and a plane in 3D?

To find the intersection between a vector and a plane in 3D, we can set the vector's position vector equal to the plane's equation and solve for the parameter t. This will give us the point where the vector intersects the plane.

## 5. How can we use vectors and planes in 3D to solve real-world problems?

Vectors and planes in 3D are useful in solving real-world problems involving motion, such as determining the trajectory of a projectile or the movement of a vehicle. They can also be used in fields such as engineering and physics to analyze forces and design structures. Additionally, 3D vectors and planes are essential in computer graphics and gaming for creating 3D models and rendering realistic scenes.

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