Final answer:
To prove the diagonals of a rectangle are equal using vectors, represent the sides of the rectangle as vectors, and then express the diagonals as vector sums. Since opposite sides of a rectangle are equal, the sums of vectors representing the diagonals are also equal, thus proving the diagonals are equal in length.
Step-by-step explanation:
To prove that the diagonals of a rectangle are equal in length using vectors, consider a rectangle with vertices A, B, C, and D. Let vector â represent the side AB, and vector Ă represent the side AD, with the rectangle situated in the xy-plane. The diagonal AC can be represented as the vector sum (A + Ă), and the diagonal BD can be represented as the vector sum (â + B). Since in a rectangle, opposite sides are equal in length, AB = CD and AD = BC, and thus â = vector CD and Ă = vector BC.
Therefore, the diagonal AC equals to A + Ă and the diagonal BD equals to â + B. Since â and Ă are equal in magnitude and direction, by the definition of equality of vectors, the diagonals AC and BD are equal. This is because they are the sums of vectors of equal magnitude and direction. Hence, the diagonals of a rectangle are equal in length when represented by vectors.
Note that the use of the Pythagorean theorem confirms this, as the sum of the squares of the components of the vectors representing the sides equals the square of the diagonal due to the rectangular properties of the shape.