Question

Using vector algebra, find the area of a parallelogram/triangle. Also, derive the area analytically and verify the same.

Answer 1

Answer:

Area of parallelogram: |a x b|

Area of traingle: ½ |ab x bc|

Step-by-step explanation:

In mathematics and physics, vector is a term that refers colloquially to some quantities that cannot be expressed by a single number, or to elements of some vector spaces.

• Area of parallelogram : base x height.

It is the region enclosed or encompassed by a parallelogram in two-dimensional space. Let us recall the definition of a parallelogram. A parallelogram is a four-sided, 2-dimensional figure with:

two equal, opposite sides,

two intersecting and non-equal diagonals, and

opposite angles that are equal

now,

let's say we have 2 vectors a and b.

then, a x b gives us vector perpendicular to plane of a and b .

so, area of parallelogram will be |a x b|.

• Area of traingle: ½ |ab x bc|

If vector AB and vector AC are given then area of triangle ABC = ½ |AB × AC|.

Using an analytical approach means you derive the formula for the area of the parallelogram, and use that formula to find the exact result.

Let's say you know the location of vertex c⃗ , and two vertices adjacent to it, a⃗ and b⃗ . You know

a⃗ =⎡⎣⎢axayaz⎤⎦⎥,b⃗ =⎡⎣⎢bxbybz⎤⎦⎥,c⃗ =⎡⎣⎢cxcycz⎤⎦⎥

To find an analytical solution to the area of the parallelogram, you need to find the function

A(ax,ay,az,bx,by,bz,cx,cy,cz)

If you are not given three vertices of the parallelogram, but say edge lengths and angles, you need to find the area of the parallelogram in terms of those variables you are given.

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Answer 2

Answer:

Area of parallelogram: |a x b|

Area of traingle: ½ |ab x bc|

Step-by-step explanation:

Area of a parallelogram is a region covered by a parallelogram in a two-dimensional plane. In Geometry, a parallelogram is a two-dimensional figure with four sides. It is a special quadrilateral case, where opposite sides are equal and parallel. The area of a parallelogram is the space enclosed within its four sides. The area of a parallelogram is equal to the product of length and height of the parallelogram.The sum of the interior angles in a quadrilateral is 360 degrees. A parallelogram has two pairs of parallel sides with equal measures. Since it is a two-dimensional figure, it has an area and perimeter.

The area of a triangle can be calculated with the help of various formulas. The basic formula that is used to find the area of a triangle is ½ × Base × Height where "Base" is the side of the triangle on which the altitude is formed, and "Height" is the length of the altitude drawn to the "Base" from its opposite vertex. However, if the altitude of a triangle is not known, and we need to find the area of triangle with 3 different sides, the Heron's formula is used. This formula was derived by a Greek mathematician known as the Heron of Alexandria.

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