A vector overrightarrow (B) , when added to vector overrightarrow (C)=3i+4j yields a resultant vector which is in the positive y direction and has a magnitude equal to that of overrightarrow (C) . What is the magnitude of overrightarrow (B)

To find the magnitude of vector $\overrightarrow{B}$, we need to determine the components of $\overrightarrow{B}$ first. Since the resultant vector is in the positive y direction and has the same magnitude as $\overrightarrow{C}$, we can write the resultant vector as $\overrightarrow{R} = 0i + 7j$. <br /><br />Using the vector addition formula, we have:<br />$\overrightarrow{R} = \overrightarrow{B} + \overrightarrow{C}$<br /><br />Substituting the given values, we get:<br />$0i + 7j = \overrightarrow{B} + (3i + 4j)$<br /><br />Simplifying the equation, we have:<br />$-3i + 3j = \overrightarrow{B}$<br /><br />Therefore, the components of $\overrightarrow{B}$ are $-3i + 3j$. <br /><br />To find the magnitude of $\overrightarrow{B}$, we use the formula magnitude of a vector:<br />$|\overrightarrow{B}| = \sqrt{(-3)^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}$<br /><br />So, the magnitude of $\overrightarrow{B}$ is $3\sqrt{2}$.