Table of Contents

## Coplanar Vectors Introduction

Coplanar Vectors: A coplanar vector is a vector that is located in the same plane as another vector. This is often done for convenience when working with vectors, as it makes calculations and manipulations simpler. In physics, a vector is a mathematical representation of a physical quantity, such as force or momentum. Vectors can be added, subtracted, and multiplied together, and they have a direction and a magnitude.

## All about Coplanar Vectors

A coplanar vector is a vector that lies in the same plane as another vector. In other words, the two vectors are parallel to each other. This can easily visualized using a two-dimensional plane. If two vectors are drawn on the plane, and they lie in the same line, then they are coplanar vectors.

The magnitude of a coplanar vector is simply the sum of the magnitudes of the individual vectors. The direction of a coplanar vector is the same as the direction of the individual vectors.

Coplanar vectors are often used in physics and engineering. For example, they can used to model the forces between objects.

## Equality of Vectors

Two vectors said to equal if they have the same magnitude and direction.

## Types of Vectors

There are three types of vectors: magnitude, direction, and displacement vectors.

- A magnitude vector is simply the magnitude (or size) of a vector. It has no direction information.
- A direction vector has both magnitude and direction information. It points in a specific direction.
- A displacement vector has magnitude and direction information, and it also has a starting point and an ending point. It shows the displacement between the two points.

## Coplanarity in Theory and Practice

### Theory

In theory, coplanarity is the condition of being on the same plane. Therefore this means that all of the features on a part are at the same height and that there are no gaps or overlaps between them.

### Practice

In practice, coplanarity is often difficult to achieve. This is because it is often difficult to measure and to maintain the same height across a variety of features. Even slight variations can cause problems with fit and assembly.

## Conditions for Coplanar Vectors/ Properties of Coplanar Vectors

Two vectors said to be coplanar if they lie in the same plane.

If vectors A and B are coplanar, then they have the following properties:

- Their sum is a vector in the same plane as A and B.
- difference is a vector in the same plane as A and B.
- Their product is a vector in the same plane as A and B.

## What are Linearly Independent Vectors?

Linearly independent vectors are vectors that can written as a linear combination of each other, but not all linear combinations of vectors are independent.

## What are Linearly Dependent Vectors?

Linearly dependent vectors are vectors that related to each other in a way that one vector can expressed as a combination of the other vectors.

## How do you Know if Two Vectors are Coplanar?

If two vectors are coplanar, then they will have the same orientation and lie in the same plane.

## FAQs

Q: What are coplanar vectors?

A: Coplanar vectors are vectors that lie in the same plane.

Q: How can you tell if two vectors are coplanar?

A: Two vectors are coplanar if there exists a plane that contains both of them.

Q: What is the cross product of two coplanar vectors?

A: The cross product of two coplanar vectors is a vector perpendicular to the plane in which they lie.

Q: Can three non-collinear vectors be coplanar?

A: No, three non-collinear vectors cannot be coplanar because they span a three-dimensional space.

Q: What is the scalar triple product of three coplanar vectors?

A: The scalar triple product of three coplanar vectors is zero, since the three vectors lie in the same plane.

Q: How many coplanar vectors are needed to define a plane?

A: Two non-collinear coplanar vectors are needed to define a plane.

Q: What is the angle between two coplanar vectors?

A: The angle between two coplanar vectors is the angle between the two lines they define in the plane they lie in.

Q: Can two collinear vectors be coplanar?

A: Yes, two collinear vectors are coplanar since they lie on the same line which can be considered as a plane.

Q: What is the projection of one coplanar vector onto another?

A: The projection of one coplanar vector onto another is the component of the first vector that lies in the direction of the second vector.