Using Vector Algebra, find the area of a parallelogram/triangle.

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

Introduction:

Vector algebra is a branch of mathematics that deals with vectors, which are mathematical objects that have both magnitude and direction. It is an essential topic of algebra that studies the algebra of vector quantities. This topic is important in physics, engineering, and computer science. One of the applications of vector algebra is finding the area of a parallelogram or triangle using vector algebra. To find the area of a parallelogram or triangle using vector algebra, we can use the cross product of two adjacent sides of the parallelogram or two sides of the triangle.

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Finding the area of a parallelogram/triangle using Vector Algebra:

To find the area of a parallelogram or triangle using vector algebra, we can use the cross product of two adjacent sides of the parallelogram or two sides of the triangle. The magnitude of the cross product gives us the area of the parallelogram or triangle.

• Area of a Parallelogram

If we have two adjacent sides of a parallelogram given by vectors a→ and b→ , then the area of the parallelogram is given by the magnitude of the cross product of a→ and b→ , i.e.,

Area of a Parallelogram = |a→ × b→|

To verify the area analytically, we can use the formula for the area of a parallelogram and calculate the area using the given coordinates of the vertices.

Example:

Let's take an example to find the area of a parallelogram using vector algebra. Suppose we have two adjacent sides of a parallelogram given by vectors a→ = ( 1 , 2 , 3 ) and b→ = ( 4, 5, 6 ). Then the area of the parallelogram is given by

To verify the area analytically, we can use the formula for the area of a parallelogram, which is given by the magnitude of the cross product of the adjacent sides. We have already calculated the cross product, which is ( -3 , 6, -3 ). Therefore, the magnitude of the cross product is 3√3, which is the same as the area we calculated using vector algebra.

• Area of a Triangle

Three vertices of a triangle given by the position vectors a→ = ( 1 , 2 , 3 ), b→ = ( 2 , -1 , 4 ), and c→ = ( 4 , 5 , -1 ). Then two sides of the triangle are given by vectors a→ - b→ and a→ - c→. Then the area of the triangle is given by:

To verify the area analytically, we can use the formula for the area of a triangle, which is given by half the magnitude of the cross product of the two sides. We have already calculated the cross product, which is ( -20 , 14 , 11 ). Therefore, the magnitude of the cross product is √757, and half of it is ½√757, which is the same as the area we calculated using vector algebra.

Conclusion:

This project on finding the area of parallelograms and triangles using vector algebra has provided me with valuable insights. By employing the cross product and magnitude calculations, I have discovered a systematic method to determine the area of these geometric shapes. Through analytical derivation and verification, I have confirmed the validity of the formulas. This project has enhanced my understanding of vector algebra and its practical applications in geometry, fostering my problem-solving skills and mathematical reasoning abilities.