Section 11.1 : Basic Concepts
5. The vector \(\vec v = \left\langle {6, - 4,0} \right\rangle \) starts at the point \(P = \left( { - 2,5, - 1} \right)\). At what point does the vector end?
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Start SolutionTo answer this problem we just need to recall that the components of the vector are always the coordinates of the ending point minus the coordinates of the starting point.
So, if the ending point of the vector is given by \(Q = \left( {{x_2},{y_2},{z_2}} \right)\) then we know that the vector \(\vec v\) can be written as,
\[\vec v = \overrightarrow {PQ} = \left\langle {{x_2} + 2,{y_2} - 5,{z_2} + 1} \right\rangle \] Show Step 2But we also know just what the components of \(\vec v\) are so we can set the vector from Step 1 above equal to what we know \(\vec v\) is. Doing this gives,
\[\left\langle {{x_2} + 2,{y_2} - 5,{z_2} + 1} \right\rangle = \left\langle {6, - 4,0} \right\rangle \] Show Step 3Now, if two vectors are equal the corresponding components must be equal. Or,
\[\begin{align*}{x_2} + 2 & = 6 & \hspace{0.1in} \Rightarrow \hspace{0.5in}{x_2} & = 4\\ {y_2} - 5 & = - 4\hspace{0.1in} & \Rightarrow \hspace{0.5in}{y_2} & = 1\\ {z_2} + 1 & = 0 & \hspace{0.1in} \Rightarrow \hspace{0.5in}{z_2} & = - 1\end{align*}\]As noted above this results in three very simple equations that we can solve to determine the coordinates of the ending point.
The endpoint of the vector is then,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{Q = \left( {4,1, - 1} \right)}}\]