Pythagorean Theorem

Definition

The Pythagorean Theorem is a fundamental relationship in geometry that describes the relationship between the sides of a right triangle. It states that the square of the length of the hypotenuse (the longest side) of a right triangle is equal to the sum of the squares of the lengths of the other two sides.

5 Must Know Facts For Your Next Test

1. The Pythagorean Theorem is used to calculate the length of the third side of a right triangle given the lengths of the other two sides.
2. In a right triangle with sides $a$, $b$, and $c$ (where $c$ is the hypotenuse), the Pythagorean Theorem states that $a^2 + b^2 = c^2$.
3. The Pythagorean Theorem is fundamental to understanding the relationships between vectors in the plane and in three dimensions.
4. The Pythagorean Theorem can be used to calculate the magnitude (length) of a vector in the Cartesian coordinate system.
5. The Pythagorean Theorem is also used to find the distance between two points in the Cartesian coordinate system.

Review Questions

• Explain how the Pythagorean Theorem can be used to calculate the magnitude of a vector in the plane.
  • The Pythagorean Theorem can be used to calculate the magnitude (length) of a vector in the Cartesian coordinate system. If a vector has components $a$ and $b$ in the $x$ and $y$ directions, respectively, then the magnitude of the vector is given by the formula $\sqrt{a^2 + b^2}$, which is derived directly from the Pythagorean Theorem. This relationship allows us to determine the length of a vector in the plane using only the vector's components.
• Describe how the Pythagorean Theorem can be used to find the distance between two points in three-dimensional space.
  • In three-dimensional space, the distance between two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ can be calculated using the Pythagorean Theorem. The distance formula is given by $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$, which represents the length of the hypotenuse of the right triangle formed by the differences in the $x$, $y$, and $z$ coordinates of the two points. This allows us to determine the spatial separation between any two points in three-dimensional space.
• Analyze how the Pythagorean Theorem is used to define the dot product of two vectors in three dimensions and explain the geometric interpretation of the dot product.
  • The dot product of two vectors $\vec{u} = (u_1, u_2, u_3)$ and $\vec{v} = (v_1, v_2, v_3)$ in three dimensions is defined as $\vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 + u_3v_3$. This definition is closely related to the Pythagorean Theorem, as the dot product can be interpreted geometrically as the product of the magnitudes of the two vectors and the cosine of the angle between them: $\vec{u} \cdot \vec{v} = \|\vec{u}\| \|\vec{v}\| \cos(\theta)$. This relationship allows us to use the Pythagorean Theorem to analyze the relative orientation and magnitudes of vectors in three-dimensional space.

Related terms

Right Triangle: A triangle in which one of the angles is a 90-degree angle, or a right angle.

Hypotenuse: The longest side of a right triangle, which is opposite the right angle.

Cartesian Coordinate System: A two-dimensional or three-dimensional coordinate system that uses perpendicular axes to specify the location of a point.