There are two types of mathematical quantities that are used to explain the nature of things in term of direction and magnitude, and they are vector or a scalar.
Definition of Vector
Vectors are geometrical entities with magnitude and direction. The length of a vector signifies its magnitude, and it is depicted as a line with an arrow pointing towards the vector’s direction. Vectors are used to express physical quantities like displacement, velocity, and acceleration. In addition, the invention of electromagnetic induction in the late nineteenth century ushered in the use of vectors. Two vectors are same if their magnitude and direction are the same. This means that if we translate a vector to a new point (without rotating it), the vector we finish up with is the same as the one we started with. Force vectors and velocity vectors are two instances of vectors. Both velocity and force are travelling in the same direction. The force’s intensity or the velocity’s matching speed would be represented by the vector’s magnitude.
Applications of Vector Analysis:
Air traffic controllers use vectors to track flights, meteorologists use them to describe wind conditions, and computer programmers use them to create virtual worlds. In this lesson, we’ll go over three vector applications that are often utilized in physics: work, torque, and magnetic force.
When a force is applied to an item or system, the term work is used to describe the energy that is added to or withdrawn from it. Work is maximum when the applied force is parallel to the motion of the item, and no work is done when the force is applied perpendicular to the motion, according to experiment. As a result, the dot product of the force vector and the displacement vector may be used to characterize the work done by a force.
When a charged particle moves perpendicular to the magnetic field, the magnetic force on the particle is greatest, and when the particle moves parallel to the field, the magnetic force is zero. As a result, the magnetic force may be expressed as the cross-product of the field strength vector and the velocity vector of the particle: F=qv x B, where F is the force acting on the particle, q is the particle’s charge, v is the particle’s velocity, and B is the magnetic field’s vector. The force will be measured in newtons, the metric base-unit of force, if the velocity is measured in m/s and the magnetic field is measured in tesla.
When you raise a baseball off a tabletop, you are exerting a force on the entire thing. When you press down on a doorknob, the door will spin on its hinges. Torque is a word used by scientists to define the force-like quality that impacts an object’s rotation. The torque may be expressed as the cross-product of the force vector and the lever arm, a radially outward vector pointing from the axis of rotation to the point where the force is applied to the object: T =r x F, where T is the torque, r is the lever arm, and F is the applied force.
Vectors are used to describe rotational motion:
In physics, we frequently need to explain rotating motion. If an item rotates, the following must be specified:
- The spinning axis around which the thing revolves
- The rotational direction of the item around that axis.
- The rotational speed of the item
To explain this form of rotating motion, we develop a new type of vector called a “axial vector.” The direction of the vector is chosen to be co-linear with the axis of rotation, and the magnitude of the vector is chosen to indicate the rotational speed of the item. As a result, we only have two options for vector direction. Consider the wheels of a car driving away from you. Because the axis of rotation is the axis of the wheel, we know that the angular velocity vector (which describes the wheel’s rotation) must point to the left or right.
Another right-hand rule is used to determine the vector’s direction. To distinguish it from the right hand rule for the cross product, we’ll call it “the right hand rule for axial vectors.” When you curl your fingers in the direction of rotation and use the right hand rule for axial vectors, the vector points in the direction of your thumb. The wheels of the automobile travelling away from you will turn so that the closest point to you moves up and the furthest point moves down. The rotation vector points to the left, according to the right hand rule.
Vector in quantum mechanics:
It is simple to emphasise the importance of linear algebra for physicists because Quantum Mechanics is totally reliant on it. Time domain (state space) control theory and tensor stresses in materials are also relevant. In quantum physics, the state of a physical system is represented as a vector in a complex vector space. The state space of the system is defined by this vector. The symbol | ⟩, sometimes known as a ket, represents such a physical state of a quantum system. This form is known as the Dirac notation, and it is widely used in quantum physics. A ket can alternatively be called a state vector, ket vector, or just state.
A vector is a number that has both magnitude and direction associated with it. Vectors are useful in a variety of situations, including those that need force or velocity. In physics, vectors are commonly employed to determine displacement, velocity, and acceleration. Vectors are arrows that indicate a magnitude and direction combination.