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# Find the components of the vector v}with given initial point P and terminal point Q. Find |v|. Sketch |v|. Find the unit vector in the direction of v.$P:(1,0,1.2), \quad Q:(0,0,6.2)$

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It \textbf v is a vector with the initial point $P:\ (x_1, y_1, z_1)$ and the terminal point $Q:\ (x_2, y_2, z_2)$, then its components are given by

$v_1=x_2-x_1,\quad v_2=y_2-y_1,\quad v_3=z_2-z_1,$

and we denote this vector by $\textbf v=[v_1, v_2,v_3]$. So the components of the vector with the given initial and terminal point $P$ and $Q$ are

\begin{align*} &v_1=0-1,\quad v_2=0-0,\quad v_3=6.2-1.2\\ \Longrightarrow\quad &\boxed{v_1=-1,\quad v_2=0,\quad v_3=5}\ . \end{align*}

The length of the vector $\textbf v=[v_1,v_2,v_3]$ is given by

$|\textbf v|=\sqrt{v_1^2+v_2^2+v_3^2},$

so the length of our vector is

\begin{align*} &|\textbf v|=\sqrt{(-1)^2+0^2+5^2}\\ \Longrightarrow\quad &\boxed{|\textbf v|=\sqrt{26}}\ . \end{align*}

We can obtain the unit vector in the direction of \textbf v by dividing the vector \textbf v with its norm (length).

\begin{align*} &\textbf x=\frac{\textbf v}{|\textbf v|}=\frac{[-1,0,5]}{\sqrt{26}}\\ \Longrightarrow\quad & \boxed{\textbf x=\left[-1/\sqrt{26},0,5/\sqrt{26}\right]}\ . \end{align*}

The sketch of the vector \textbf v is given in the figure below.

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