A vector has magnitude and direction and two vectors with same magnitude and direction regardless of positions of their initial points are always equal.

*“Mathematics is the
science of the connection of magnitudes.
Magnitude is anything that can be put equal or unequal to another
thing. Two things are equal when in
every assertion each may be replaced by the other.” * - Hermann Günther Grassmann

We are familiar with the concept of vectors, (** vectus **in
Latin means “to carry”) from our XI standard text book. Further the modern
version of Theory of Vectors arises from the ideas of Wessel(1745-1818) and
Argand (1768-1822) when they attempt to describe the complex numbers
geometrically as a directed line segment in a coordinate plane. We have seen
that

We also have studied addition of two vectors, scalar
multiplication of vectors, dot product, and cross product by denoting an
arbitrary vector by the notation * *or a_{1}ˆ*i* + a_{2}ˆ*j* + a_{3}ˆ*k*.

To understand the direction and magnitude of a given
vector and all other concepts with a little more rigor, we shall recall the
geometric introduction of vectors, which will be useful to discuss the
equations of straight lines and planes. Great mathematicians Grassmann,
Hamilton, Clifford and Gibbs were pioneers to introduce the dot and cross
products of vectors.

The vector algebra has a few direct applications in physics and
it has a lot of applications along with vector calculus in physics,
engineering, and medicine. Some of them are mentioned below.

·
To calculate the volume of a parallelepiped, the scalar triple
product is used.

·
To find the work done and torque in mechanics, the dot and cross
products are respectiveluy used.

·
To introduce curl and divergence of vectors, vector algebra is
used along with calculus. Curl and divergence are very much used in the study
of electromagnetism, hydrodynamics, blood flow, rocket launching, and the path
of a satellite.

·
To calculate the distance between two aircrafts in the space and
the angle between their paths, the dot and cross products are used.

·
To install the solar panels by carefully considering the tilt of
the roof, and the direction of the Sun so that it generates more solar power, a
simple application of dot product of vectors is used. One can calculate the
amount of solar power generated by a solar panel by using vector algebra.

·
To measure angles and distance between the panels in the
satellites, in the construction of networks of pipes in various industries,
and, in calculating angles and distance between beams and structures in civil
engineering, vector algebra is used.

Upon completion of this chapter, students will
be able to

● apply
scalar and vector products of two and three vectors

● solve
problems in geometry, trigonometry, and physics

● derive
equations of a line in parametric, non-parametric, and cartesian forms in
different situations

● derive
equations of a plane in parametric, non-parametric, and cartesian forms in different
situations

● find
angle between the lines and distance between skew lines

● find
the coordinates of the image of a point

Tags : Applications of Vector Algebra , 12th Mathematics : UNIT 6 : Applications of Vector Algebra

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12th Mathematics : UNIT 6 : Applications of Vector Algebra : Introduction | Applications of Vector Algebra

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