Vector

Definition

A vector is a mathematical quantity that has both magnitude and direction. It is used to represent physical quantities such as force, velocity, and displacement, which require both a size and a direction to be fully described.

5 Must Know Facts For Your Next Test

1. Vectors can be represented graphically as arrows, where the length of the arrow represents the magnitude and the direction of the arrow represents the direction of the vector.
2. Vectors in two-dimensional space can be described using an $x$-component and a $y$-component, while vectors in three-dimensional space can be described using an $x$-component, a $y$-component, and a $z$-component.
3. The magnitude of a vector is the length or size of the vector, and it is calculated using the Pythagorean theorem.
4. Vectors can be added or subtracted using the parallelogram rule or the head-to-tail method, which allows for the combination of multiple vectors.
5. Vectors are essential in the study of physics, engineering, and many other scientific fields, as they are used to describe and analyze quantities that have both magnitude and direction.

Review Questions

• Explain how vectors can be used to represent physical quantities in the plane.
  • Vectors in the plane can be used to represent physical quantities such as displacement, velocity, and force. The $x$-component of the vector represents the horizontal part of the quantity, while the $y$-component represents the vertical part. The magnitude of the vector corresponds to the size or magnitude of the physical quantity, and the direction of the vector corresponds to the direction of the physical quantity.
• Describe how vectors can be used to represent physical quantities in three-dimensional space.
  • In three-dimensional space, vectors can be used to represent physical quantities such as displacement, velocity, and force by incorporating a $z$-component in addition to the $x$- and $y$-components. The magnitude of the vector corresponds to the size or magnitude of the physical quantity, and the direction of the vector corresponds to the direction of the physical quantity in three-dimensional space. This allows for the complete description of physical quantities that have both magnitude and direction in all three spatial dimensions.
• Explain how the concept of vectors is used in the equations of lines and planes in space.
  • The concept of vectors is essential in the equations of lines and planes in three-dimensional space. The direction of a line can be represented by a vector, and the equation of the line can be expressed in vector form. Similarly, the normal vector to a plane, which is perpendicular to the plane, can be used to represent the orientation of the plane, and the equation of the plane can be expressed in vector form. Understanding the role of vectors in these equations allows for the efficient and accurate description of the geometry of lines and planes in three-dimensional space.

Related terms

Scalar:

A scalar is a mathematical quantity that has only a magnitude, without a specific direction. Examples include temperature, mass, and time.

Unit Vector: A unit vector is a vector with a magnitude of 1, which is used to represent the direction of another vector.

Vector Addition: Vector addition is the process of combining two or more vectors to find a single vector that has the same effect as the original vectors.