If the vector $A$ has a magnitude of 3 , and vector $B$ has a magnitude of 4 , and their directio right angles to each other, what is the magnitude of vector $C$, where $C=A+B$ ?

Step 1 ：Given that the vector $A$ has a magnitude of 3, and vector $B$ has a magnitude of 4, and they are at right angles to each other.

Step 2 ：We are asked to find the magnitude of vector $C$, where $C=A+B$.

Step 3 ：The magnitude of the resultant vector of two vectors at right angles to each other can be found using the Pythagorean theorem. The magnitude of vector $C$ is given by the square root of the sum of the squares of the magnitudes of vectors $A$ and $B$.

Step 4 ：Substituting the given values into the formula, we get \(C = \sqrt{A^2 + B^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25}\).

Step 5 ：Calculating the square root of 25, we get $C = 5$.

Step 6 ：Final Answer: The magnitude of vector $C$ is \(\boxed{5}\).