
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:
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 6cm2
and 8cm2, the total areas of these two rectangles is (6 + 8)cm2
i.e. 14cm2.
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,
a1x + b1y = c1
………………. (i)
a2x + b2y = c2
………………. (ii)
Writing equations (i) and (ii) in the
matrix form, we get
Equating the corresponding elements of equal
matrices X and A-1B, 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,
1st case, 17x + 4y = 1110 …………… (i)
2nd 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 aij
is the element in ith row and jth column of A. Then the minor of the element aij is
the determinant of the matrix formed by omitting ith row and jth
column of A. The minor of element aij
is denoted by Mij.
Let A be a square matrix, and Aij be the cofactors of the elements aij 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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