Locate the point \(P~(3,5,4)\) and sketch the position vector \(\overrightarrow{OP} = 3\hat{i}+5\hat{j}+4\hat{k}\), where \(\hat{i}=(1,0,0), \hat{j}=(0,1,0)\), and \(\hat{k}=(0,0,1)\) are the special unit vectors pointing in the direction of the positive \(x\)-, \(y\)-, and \(z\)-axes, respectively. Calculate the magnitude of \(\overrightarrow{OP}\).

Solution

Note: The \(x\) and \(y\) coordinates help form the bottom (if \(z\) is positive) or the top (if \(z\) is negative) of the rectangular prism.

We see \(\lvert {\color{NavyBlue}\overrightarrow{OQ}}\rvert = \sqrt{3^2+5^2} =\sqrt{34}\).

Thus, by the Pythagorean theorem

Observe \(\lvert {\color{NavyBlue}\overrightarrow{OQ}}\rvert^2 = 3^2+5^2=34\) and \(\lvert \overrightarrow{PQ}\rvert^2=4^2=16\), then

In general, if \(\overrightarrow{u}=(a,b,c)\), then \(\lvert \overrightarrow{u}\rvert = \sqrt{a^2+b^2+c^2}\).