An abstract representation of the mathematical concept of two vectors u and v fitting the equation (u - v) * (u - v) = u * u + v * v. The vectors should be depicted as two arrows originating from the same point, pointing in different directions. The concept that is represented by this equation, the Pythagorean theorem, should be suggested visually, let's say through the use of subtly incorporated right-angled triangles. No text should be present in the image.

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?

(u − v) • (u − v) = u•u + v•v -2u•v

Is what you get by multiplying out the dot products.

For the equation
(u − v) • (u − v) = u•u + v•v

To be valid, u•v must be zero, so u and v must be perpendicular. In that case the triangle formed by u, v and the hypotenuse u -v is a right triangle and the left side is the square of the hypotenuse, as required by the Pythagorian theorem.

Well, to figure out how these vectors must be related, let's expand the equation and take it step by step.

(u - v) • (u - v) = u • u + v • v

Expanding the left side, we get:

u • u - 2u • v + v • v = u • u + v • v

Simplifying, we see that the u • u and v • v terms cancel out on both sides:

-2u • v = 0

Dividing by -2, we get:

u • v = 0

So, u and v must be orthogonal or perpendicular to each other for this equation to hold.

As for the familiar theorem, this equation represents the Pythagorean theorem in vector form. It states that the sum of the squares of the lengths of the two sides of a right triangle is equal to the square of the length of the hypotenuse. In this case, the hypotenuse is represented by the vector u - v, and the other two sides are represented by u and v.

To determine the relationship between vectors u and v, let's first expand the equation (u-v) • (u-v) = u • u + v • v.

Expanding the left side of the equation yields:
(u - v) • (u - v) = u • u - u • v - v • u + v • v.

Simplifying, we can rewrite the equation as:
u • u - 2u • v + v • v = u • u + v • v.

Canceling out the common terms on both sides, we get:
-2u • v = 0.

To solve for the relationship between u and v, divide both sides by -2:
u • v = 0.

Therefore, the dot product (also called the scalar product) of vectors u and v must be zero. This means that u and v are orthogonal, or perpendicular to each other.

The equation (u − v) • (u − v) = u•u+v•v represents the Pythagorean theorem in vector form, where the sum of the squares of the lengths of the sides (u - v) and (u - v) is equal to the sum of the squares of the lengths of the sides u and v.

To determine how the vectors u and v must be related to satisfy the equation (u - v) • (u - v) = u • u + v • v, we can expand the left-hand side (LHS) and right-hand side (RHS) of the equation separately and analyze the result.

Expanding the LHS:
(u - v) • (u - v) = (u • u) - (u • v) - (v • u) + (v • v)

Expanding the RHS:
u • u + v • v

Since the LHS and RHS are equal, we can equate the corresponding components:

(u • u) - (u • v) - (v • u) + (v • v) = u • u + v • v

By rearranging terms, we get:

- (u • v) - (v • u) = 0

The dot product of two vectors is commutative, which means u • v = v • u. Applying this property to the equation, we can simplify it further:

-2(u • v) = 0

To solve this equation, we divide both sides by -2:

u • v = 0

Therefore, the two vectors u and v must have a dot product of zero, meaning they are orthogonal or perpendicular to each other.

The equation (u - v) • (u - v) = u • u + v • v represents the Pythagorean theorem in vector form. It states that in a two-dimensional space, the square of the hypotenuse (the difference of the two vectors) is equal to the sum of the squares of the other two sides (the individual vectors squared).