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Improved visualization by replacing words with symbols and added line breaks (#23686)

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* Improved visualization

Replaced theta to actual symbols for better visualization, and added line breaks where they should be

* Improved formatting
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Emily Gong
2018-11-29 14:07:54 -08:00
committed by Manish Giri
parent 1c5cc45bb4
commit cd211bcf5e
1 changed files with 11 additions and 6 deletions
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17
guide/english/mathematics/2-by-2-determinants/index.md
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@ -15,12 +15,17 @@ Given a matrix A, the determinant of A (written as |A|) is given by the followin
## Properties of (2x2) determinants ## Properties of (2x2) determinants
The rows and vectors of a 2 by 2 matrix can be associated with points on a cartesian plane, such that each row forms a 2D vector. These two vectors form a parallelogram, as shown in the image below. The rows and vectors of a 2 by 2 matrix can be associated with points on a cartesian plane, such that each row forms a 2D vector. These two vectors form a parallelogram, as shown in the image below.
PROOF:
Let the vectors be M(a,b),N(c,d) originating from origin in a 2-D plane with an angle (*theta*>0) between them(head of one vector touching tail of another vector). But in here it doesn't matter because sin(theta)=sin(2(pi)-theta). Then the other point is P(a+c,b+d). The area of the parallelogram is perpendicular distance from one point say N(c,d) to the base vector, M(a,b) multiplied by the length of the base vector, |M(a,b)|. The parallelogram consists of two triangles hence, the area is two times of a triangle. ### Proof
Let the perpendicular distance be h
h=|N(c,d)|* sin(*theta*(angle between two vectors)) Let the vectors be M(a,b),N(c,d) originating from origin in a 2-D plane with an angle (θ>0) between them (head of one vector aligning with tail of another vector). But in here it doesn't matter because sin(θ)=sin(2π-θ). Then the other point is P(a+c,b+d). The area of the parallelogram is perpendicular distance from one point say N(c,d) to the base vector, M(a,b) multiplied by the length of the base vector, |M(a,b)|. The parallelogram consists of two triangles hence, the area is two times of a triangle.
b=|M(a,b)|
Area=h * b Let the perpendicular distance be h.
Then:
- h=|N(c,d)| * sin(θ)
- b=|M(a,b)|
- Area = h * b
The absolute value of the determinant is equal to the area of the parallelogram. The absolute value of the determinant is equal to the area of the parallelogram.