The Pythagorean theorem is a fundamental principle in geometry that applies to right-angled triangles. The theorem states that in a right-angled triangle, the square of the length of the hypotenuse [tex]\(c\)[/tex] is equal to the sum of the squares of the lengths of the other two sides [tex]\(a\)[/tex] and [tex]\(b\)[/tex]. Mathematically, this is expressed as:
[tex]\[ c^2 = a^2 + b^2 \][/tex]
In the context of vector addition, if two vectors [tex]\( \vec{A} \)[/tex] and [tex]\( \vec{B} \)[/tex] are perpendicular to each other, the magnitude [tex]\( R \)[/tex] of the resultant vector [tex]\( \vec{R} \)[/tex] can be found using the same principle. The resultant vector [tex]\( \vec{R} \)[/tex] is the hypotenuse of a right-angled triangle formed by the vectors [tex]\( \vec{A} \)[/tex] and [tex]\( \vec{B} \)[/tex].
To find the magnitude [tex]\( R \)[/tex] of the resultant vector [tex]\( \vec{R} \)[/tex], we use:
[tex]\[ R^2 = A^2 + B^2 \][/tex]
where [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are the magnitudes of vectors [tex]\( \vec{A} \)[/tex] and [tex]\( \vec{B} \)[/tex], respectively.
Therefore, the equation that represents the Pythagorean theorem, which can be used to find the magnitude of the resultant vector when [tex]\( \vec{A} \)[/tex] and [tex]\( \vec{B} \)[/tex] are perpendicular, is:
[tex]\[ R^2 = A^2 + B^2 \][/tex]
So, the correct equation among the given options is:
[tex]\[ R^2 = A^2 + B^2 \][/tex]
