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I always have a hard time visualizing how this is done.

I know, given two points $p$ and $q$, the vector $\vec{pq}$ in $\mathbb{R}^3=\langle q_x-p_x,q_y-p_y,q_z-p_z\rangle$.

But if I am given a vector and one of either $p$ or $q$? How do I calculate the other one?

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up vote 2 down vote accepted

I know, given two points $p$ and $q$, the vector $\vec{pq}$ in $\mathbb{R}^3=\langle q_x-p_x,q_y-p_y,q_z-p_z\rangle$.

But if I am given a vector and one of either $p$ or $q$? How do I calculate the other one?

We can think of $\vec p, \vec q$ as being the vectors corresponding to points $p, q$ respectively, i.e. vectors from the origin to the coordinates represented by point $p$, $q$.

Then given vector $\vec v = \vec{pq} = \vec q- \vec p$, and say, point $p = (p_x, p_y, p_z) \iff \vec p = \langle p_x, p_y, p_z \rangle$, then $$\vec q = \vec{pq} + \vec p$$

So, we are adding two vectors, $\vec{pq} + \vec p$ using vector arithmetic, resulting in $\vec q$, where point $q = (q_x, q_y, q_z)$ is given by reading off the coordinates resulting from the addition.

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A shame no feedback at all from OP! +1 –  Amzoti Jun 4 '13 at 1:18
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Use the fact that $\vec{pq} = q-p$ and that vector subtraction is defined component wise.

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So $\vec{pq}+p=q$? How does that work by adding a vector and a point? –  agent154 Jun 3 '13 at 15:18
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Note that $p$ can be interpreted as a vector from the origin to the coordinates represented by $p$. So, we are doing vector arithmetic. –  response Jun 3 '13 at 15:19
    
Oh, ok... I see. This helps a lot then to remember how to solve for points. Thanks. –  agent154 Jun 3 '13 at 15:20
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