To identify which equation represents the Pythagorean theorem, we need to recall what the Pythagorean theorem states. The Pythagorean theorem is typically used in the context of right-angled triangles and relates the lengths of the sides of the triangle. Specifically, it states:
[tex]\[ c^2 = a^2 + b^2 \][/tex]
where [tex]\(c\)[/tex] is the length of the hypotenuse (the side opposite the right angle), and [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are the lengths of the other two sides.
Translating this into the context of vectors, if we consider [tex]\(A\)[/tex] and [tex]\(B\)[/tex] as vector magnitudes and [tex]\(R\)[/tex] as the resultant vector, the Pythagorean theorem helps us find the magnitude of the resultant vector when the vectors are perpendicular to each other.
Given the options:
1. [tex]\( R = A + B \)[/tex]
2. [tex]\( R = A \times B \)[/tex]
3. [tex]\( R^2 = A^2 \times B^2 \)[/tex]
4. [tex]\( R^2 = A^2 + B^2 \)[/tex]
We compare these to the structure of the Pythagorean theorem.
1. [tex]\( R = A + B \)[/tex] suggests a direct sum of the magnitudes, which does not match the Pythagorean theorem structure.
2. [tex]\( R = A \times B \)[/tex] suggests a product of the magnitudes, which is also not related to the Pythagorean theorem.
3. [tex]\( R^2 = A^2 \times B^2 \)[/tex] suggests the squares of the magnitudes being multiplied, which is also inconsistent with the Pythagorean theorem representation.
4. [tex]\( R^2 = A^2 + B^2 \)[/tex] matches the form of the Pythagorean theorem exactly, as it shows the sum of the squares of the magnitudes equating to the square of the resultant magnitude.
Thus, the correct equation that represents the Pythagorean theorem and can be used to find the magnitude of resultant vectors is:
[tex]\[ R^2 = A^2 + B^2 \][/tex]
Therefore, the correct choice is the fourth equation.
