Calculate the area of the triangle determined by the two...

Calculate the area of the triangle determined by the two vectors $\vec{A}=3 \hat{i}+4 \hat{j}$ and $\vec{B}=-3 \hat{i}+7 \hat{j}$
(A) $\frac{33}{2}$ sq. unit
(B) $\frac{35}{2} \mathrm{sq}$. unit
(C) $\frac{45}{2}$ sq. unit
(D) 15 sq. unit

Key Concepts

Area of a Triangle using Vectors

The area of a triangle formed by two vectors can be determined by taking half of the area of the parallelogram spanned by the vectors. This is calculated by finding the magnitude of the cross product of the vectors (or using the determinant method in 2D), and then dividing the result by two. This concept is crucial in transforming vector information into geometric area measurements.

Determinant Method in 2D

In two-dimensional space, the area of the parallelogram formed by two vectors can be computed using the determinant of a 2x2 matrix that contains the vectors' components. The absolute value of this determinant gives the area, and it provides a straightforward method for area calculation without the need for three-dimensional vector operations.

Vectors and Their Components

Vectors are mathematical objects that have both magnitude and direction. In the context of geometry and physics, vectors in the plane are represented by their components along the x and y axes. This representation allows for operations like addition, subtraction, and scaling, and is fundamental in solving problems involving directions and magnitudes in two dimensions.

Cross Product and Its Magnitude

The cross product is an operation that, when applied to two vectors, yields a vector whose magnitude is proportional to the area of the parallelogram spanned by the original vectors. In two dimensions, the magnitude of the cross product is used to find areas, and its absolute value is equivalent to the determinant of a 2x2 matrix formed by the vectors’ components.

Transcript

Please give Ace some feedback