Vector

Definition

A vector is a mathematical quantity that has both magnitude (size or length) and direction. Vectors are used to represent physical quantities, such as force, velocity, and acceleration, that have a direction as well as a size.

5 Must Know Facts For Your Next Test

1. Vectors can be represented geometrically as arrows, with the length of the arrow representing the magnitude and the direction of the arrow representing the direction of the vector.
2. Vectors can be added and subtracted using the rules of vector addition and subtraction, which involve the parallelogram law or the triangle method.
3. The dot product and cross product are two important operations that can be performed on vectors, with the dot product resulting in a scalar and the cross product resulting in a new vector.
4. Vectors can be used to represent physical quantities in two-dimensional and three-dimensional space, and are fundamental to many areas of physics and mathematics.
5. The components of a vector, which are the projections of the vector onto the coordinate axes, can be used to represent the vector in a coordinate system.

Review Questions

• Explain how vectors differ from scalars and describe the key characteristics of vectors.
  • Vectors are mathematical quantities that have both magnitude (size or length) and direction, whereas scalars have only magnitude and no direction. The key characteristics of vectors are that they can be represented geometrically as arrows, with the length of the arrow representing the magnitude and the direction of the arrow representing the direction of the vector. Vectors can be added, subtracted, and combined using various operations, such as the dot product and cross product, which are fundamental to many areas of physics and mathematics.
• Explain the concept of vector addition and subtraction, and describe how these operations can be performed using the parallelogram law or the triangle method.
  • Vector addition and subtraction are important operations that can be performed on vectors. To add two vectors, the parallelogram law states that the sum of the two vectors is represented by the diagonal of the parallelogram formed by the two vectors. Alternatively, the triangle method involves placing the two vectors head-to-tail and drawing a third vector from the tail of the first vector to the head of the second vector, which represents the sum of the two vectors. Vector subtraction is performed by adding the negative of the second vector to the first vector, using the same geometric principles.
• Discuss the importance of vectors in the context of physics and mathematics, and explain how the components of a vector can be used to represent the vector in a coordinate system.
  • Vectors are fundamental to many areas of physics and mathematics, as they are used to represent physical quantities that have both magnitude and direction, such as force, velocity, and acceleration. The components of a vector, which are the projections of the vector onto the coordinate axes, can be used to represent the vector in a coordinate system. This allows for the vector to be expressed in terms of its x, y, and z components, which can be useful for performing calculations and analyzing the behavior of the physical quantity represented by the vector. The ability to represent vectors in a coordinate system is essential for many applications in physics, engineering, and other scientific fields.

Related terms

Scalar:

A scalar is a quantity that has only magnitude, or size, and no direction. Examples of scalars include temperature, mass, and time.

Magnitude:

The magnitude of a vector is its size or length, without regard to its direction.

Direction:

The direction of a vector is the orientation of the quantity it represents, such as the direction of a force or the direction of motion.