If two vectors u and v fit the equation (u − v) • (u − v) = u•u+v•v, how must these vectors u and v be related? What familiar theorem does this equation represent?
To understand how the vectors u and v must be related based on the equation (u - v) • (u - v) = u • u + v • v, let's break down the equation step by step.
First, let's expand the left-hand side (LHS) of the equation:
(u - v) • (u - v) = u • u + v • v
Expanding the dot product:
(u • u - 2u • v + v • v) = u • u + v • v
Rearranging the terms:
-2u • v = 0
Now, to solve this equation, we can divide both sides by -2, resulting in:
u • v = 0
This means that the dot product of vectors u and v is zero. In other words, the vectors u and v are orthogonal (perpendicular) to each other.
Now, let's discuss the familiar theorem that this equation represents. The equation (u - v) • (u - v) = u • u + v • v is derived from the Pythagorean theorem in Euclidean geometry.
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. In vector terms, if u and v are orthogonal vectors, their sum (u + v) forms a right-angled triangle. Applying the Pythagorean theorem to the length of this diagonal vector, we get:
(u + v) • (u + v) = u • u + v • v
Expanding and simplifying further, we get:
u • u + 2u • v + v • v = u • u + v • v
By canceling out the common terms on both sides, we arrive at:
2u • v = 0
Dividing both sides by 2 yields:
u • v = 0
This equation represents the Orthogonality Theorem, which states that two vectors are orthogonal (perpendicular) if and only if their dot product is zero. Therefore, the equation (u - v) • (u - v) = u • u + v • v represents the concept of vectors being orthogonal to each other.