A certain vector in the x y -plane has an x -component of 4 ~m and...

Here, $\mathbf{A}=4 \mathbf{i}+10 \mathbf{j}$$\begin{aligned} \therefore \quad|\mathrm{A}|=\sqrt{(4)^2+(10)^2} & =\sqrt{16+100} \\ & =\sqrt{116} \mathrm{~m} \end{aligned}$$$Now, according to question, So,$$\begin{aligned} \mathbf{A} & =8 \mathbf{i}+n \mathbf{j} \\ |\mathbf{A}| & =\sqrt{(8)^2+n^2} \\ \sqrt{116} & =\sqrt{(8)^2+n^2} \end{aligned}$$squaring both sides$$\begin{aligned} 116 & =64+n^2 \\ or \quad n^2 & =116-64=52 \\ or \quad n & =7.2 \mathrm{~m} \end{aligned}$$