**Vector Geometry: **We can study different properties and
relations relating to geometry with the help of vectors. Such a study is known
as the **Vector Geometry**.

Here are 12 **geometrical theorems** and
their proofs by **vector method**.

**Vector Geometry: **We can study different properties and
relations relating to geometry with the help of vectors. Such a study is known
as the **Vector Geometry**.

Here are 12 **geometrical theorems** and
their proofs by **vector method**.

**Product of Vectors: ****Product of vectors** or **multiplication of vectors** can be
performed by the following two ways:

- Scalar Product or Dot Product
- Vector Product or Cross
Product

**Vector Operations: **The vectors can be added, subtracted or
multiplied with one another. Addition of vectors, difference of vectors, multiplication
of a vector by a scalar, product of vectors, etc. are called **vector operations**.

**Types of Vectors: **After we have learnt ‘**Vectors and its Components**’ and ‘**Magnitude and Direction of a Vector**’, we’ll learn the Types of Vectors here. Vectors are of
different types according to their magnitude, direction, position with respect
to other given vector, and by the method of representing vectors.

Following are the different types of
vectors:

**Magnitude of a vector** is the
length of its directed line segment from the initial point to the terminal
point. So, the magnitude of a vector is a positive number which is the measure
of the line segment representing that vector. Magnitude of a vector is also
known as the **modulus** or the **absolute value** of the vector.

The quantities which can be measured are
called **physical quantities**. Some
physical quantities have **magnitudes**
only but some physical quantities have magnitudes as well as **directions**.

Let us study the addition of some
physical quantities. For example, if we have two rectangles of areas 6cm^{2}
and 8cm^{2}, the total areas of these two rectangles is (6 + 8)cm^{2}
i.e. 14cm^{2}.

If two forces 4N and 5N act on a body,
what is the total force acting on the body, 9N or 1N? Its answer is uncertain
unless their directions are known. If two forces have same direction the total
force on the body is 4N + 5N = 9N.

If two forces have opposite direction the
total force on the body is 5N – 4N = 1N.

In the calculation of some physical
quantities, the **direction** has very
important role.

The simultaneous equations in two
variables can be solved by different methods: substitution method, cross
multiplication method etc. Here, we deal with the method of solving linear
equations of two variables using **matrices**.

Let us consider the following linear
equations,

a_{1}x + b_{1}y = c_{1}
………………. (i)

a_{2}x + b_{2}y = c_{2}
………………. (ii)

Writing equations (i) and (ii) in the
matrix form, we get

Equating the corresponding elements of equal
matrices X and A^{-1}B, we get the solution.

*Note:** If the determinant |A| = 0, the system of simultaneous equations have no unique
solution, therefore when |A| ≠ 0, the equations have unique solution.*

*Example 3: If the cost of 17 kg sugar and 4 kg tea is Rs. 1110 and
the cost of 8 kg sugar and 2 kg tea is Rs. 540. Find the cost of sugar per kg and
tea per kg by matrix method.*

*Solution:** Let,*

*Cost of sugar per kg = Rs. x*

*Cost of tea per kg = Rs. y*

*Then according to questions,*

*1 ^{st} case, 17x + 4y = 1110 …………… (i)*

*2 ^{nd} case, 8x + 2y = 540 ……………… (ii)*

*Writing the given equations in the matrix form, we get*

*Comparing the corresponding elements, we get*

*x = 30, y = 150*

*Hence, The cost of
sugar per kg = Rs. 30*

* The cost of tea per
kg = Rs. 150*

*You can comment your questions or problems
regarding the solution of simultaneous equations by matrix method here.*

Let A be a square matrix and a_{ij}
is the element in i^{th} row and j^{th} column of A. Then the **minor** of the element a_{ij} is
the determinant of the matrix formed by omitting i^{th} row and j^{th}
column of A. The **minor** of element a_{ij}
is denoted by **M _{ij}**.

Let A be a square matrix, and A_{ij} be the cofactors of the elements
a_{ij} of the matrix A, then **adjoint**
or **adjugate** of A denoted by **adj A** is the matrix obtained by
transposing the matrix of cofactors of A.

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