# Vectors & Planes: Proving Perpendicularity to Plane Passing through Origin

In summary, the vector -; = Xl + yj + zk is called the position vector and points from the origin to an arbitrary point in space with coordinates (x, y, z). All points (x, y, z) that satisfy the equation Ax + By + Cz = 0, where A, B, and C are constants, lie in a plane that passes through the origin and is perpendicular to the vector Ai + Bj + Ck. To prove this, you can use the fact that if the dot product of two vectors is equal to zero, then they are perpendicular. In order to solve this problem, it would be helpful to have a basic understanding of linear algebra and 3-dimensional space.

## Homework Statement

The vector -; = Xl + yj + zk, called the position vector,
points from the origin (0. 0, o) to an arbitrary point in space with
coordinates (x, y, z). Use what you know about vectors to prove
the following: All points (x, y, z) that satisfy the equation
Ax + By + Cz = 0, where A, B, and Care constants,lie in a plane
that passes through the origin and that is perpendicular to the vector
Ai + Bj + ck. Sketch this vector and the plane.

## Homework Equations

if A . B = 0 then A is perpendicular to B

## The Attempt at a Solution

I think I first should find an equation for a vector that lies in this plane then if the cross product is zero then it is perpendicular but I don't know anything about linear algebra or 3 dimensional space and this is from a physics text.

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any hints? and what are the things that I should know as a prerequisite to solve such problem because I didn't study anything like it before.

## 1. What is the definition of a vector?

A vector is a mathematical object that has both magnitude (size) and direction. It is typically represented by an arrow pointing in the direction of its magnitude.

## 2. How do you prove two vectors are perpendicular?

To prove that two vectors are perpendicular, their dot product must equal zero. The dot product is calculated by multiplying the corresponding components of the two vectors and then adding them together. If the dot product is zero, the vectors are perpendicular.

## 3. How do you prove a vector is perpendicular to a plane?

To prove that a vector is perpendicular to a plane, you can use the normal vector of the plane. The normal vector is a vector that is perpendicular to all vectors in the plane. If the dot product of the given vector and the normal vector is zero, the vector is perpendicular to the plane.

## 4. What is the equation for a plane passing through the origin?

The equation for a plane passing through the origin is ax + by + cz = 0, where (a,b,c) is the normal vector of the plane. This means that any vector that is perpendicular to the normal vector will also be perpendicular to the plane.

## 5. Can a vector be perpendicular to a plane that does not pass through the origin?

Yes, a vector can be perpendicular to a plane that does not pass through the origin. In this case, the equation for the plane will be ax + by + cz = d, where d is a constant. The vector will still be perpendicular to the plane if its dot product with the normal vector (a,b,c) is zero.

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