components of vector in 1d ,2d and 3d
Answers
Explanation:
Vectors in 2D and 3D
1.1 Definition of vectors
Many times in engineering, one wants to model quantities that are not adequately described by a
single number, like temperature or pressure, but rather by a direction and magnitude. These are
called vector quantities or simply vectors. Examples of vector quantities are:
• Force: it has a magnitude, the strength, and a direction.
• Velocity: it also has a magnitude, the speed, and a direction.
• Acceleration.
• Electric and magnetic fields.
• A trajectory when moving along a straight line for some time t.
1.2 Notation
If a quantity is a vector, then it is either going to be in boldface, such as u or have an arrow over
it, such as ~u.
1.3 Geometric interpretation of a vector
A vector ~u has a direction and a magnitude. A convenient geometrical representation of vector is
a straight line segment drawn in space in the direction of the vector, with an arrowhead indicating
its sense. The length of the line segment is given by the magnitude of the vector. This is sometimes
called a directed straight line segment.
In this course, we will primarily use the Cartesian coordinate system to represent vectors analytically, that is using numbers. Let’s assume that we draw a vector ~u starting from the origin. It will
end at some point P with Cartesian coordinates (u1, u2, u3). By definition, these are the Cartesian
coordinates of vector ~u. For example, on Figure 1, the vectors ~u = (4, 3) and ~v = (2, 3, 1) are
shown. The magnitude or length of ~u is given by:
|~u| =
q
u
2
1 + u
2
2 + u
2
3