Given the vectors $\overrightarrow{u}= (1; u_2)$ and $\overrightarrow{v} = (4; -1)$, where $|\overrightarrow{u}-\overrightarrow{v}| = 5$, find the unknown coordinate $u_2$ of the vector $\overrightarrow{u}$.
Jane proceeded as follows:
(1) $\overrightarrow{u} -\overrightarrow{v}= (-3; u_2 + 1)$
(2) $|\overrightarrow{u}-\overrightarrow{v}| = \sqrt{(-3)^2+(u_2+ 1)^2} = 5$
(3) Next, Jane solved the equation:
$$\sqrt{(-3)^2+(u_2+ 1)^2} = 5$$
(a) She squared both sides of the equation:
$$9 +(u_2+ 1)^2=25$$(b) Then, she subtracted $9$ from both sides of the equation: $$(u_2+ 1)^2=16$$
(c) Next, she took the square root of both sides of the equation: $$u_2+ 1=4$$
(d) Finally, she determined $u_2$: $$u_2=3$$
Jane's solution is not correct. Where did Jane make a mistake in her procedure?
The mistake is in step (1). The difference of vectors $\overrightarrow{u}$ and $\overrightarrow{v}$ is $\overrightarrow{u}-\overrightarrow{v}=(-3; u_2-1)$.
The mistake is in step (2). The norm of the difference of $\overrightarrow{u}$ and $\overrightarrow{v}$ is $|\overrightarrow{u} -\overrightarrow{v}| = \sqrt{(-3)^2 \cdot(u_2+ 1)^2 }$.
The mistake is in step (3a). When we square both sides of the equation, we get $-9 +(u_2+ 1)^2=25$.
The mistake is in step (3c). When we take the square root of both sides of the equation, we get $|u_2+ 1|=4$
The correct solution of the equation $\sqrt{(-3)^2+(u_2+1)^2} = 5 $ is:
$$\begin{alignat}2 \sqrt{(-3)^2+(u_2+1)^2} &= 5\quad &&/^2\cr 9 +(u_2+ 1)^2&=25 &&/-9\cr (u_2+ 1)^2&=16 &&/^\sqrt{} \cr |u_2+ 1|&=4 ⇔ &&\ (u_2+ 1=4)\vee(u_2+ 1=-4 )\cr & &&\ (u_2=3)\vee(u_2=-5) \end{alignat}$$