Magnitude Of A Vector

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Magnitude of a Vector

The magnitude of the vector v = 6 i + 2 j+ 3 k

Figure 1: The magnitude of the vector v = 6i + 2j + 3k

Figure 1: The magnitude of the vector v = 6i + 2j + 3k

The magnitude of the vector v, written |v| or v, is the length of the arrow representing v. In Figure 1, the vector v = 6 i + 2 j+ 3 k is shown in blue. Its magnitude is the length OP. By Pythagoras in the triangle OBP, we have that OP^2=OB^2+BP^2. But by Pythagoras in the triangle OAB, we have that OB^2=OA^2+AB^2. Putting these together, we have that

\begin{eqnarray} OP^2&=&OA^2+AB^2+BP^2\\ &=&6^2+2^2+3^2\\ &=&49, \end{eqnarray}

and therefore that |v| = \sqrt{49} = 7.

In general, the magnitude of the vector a_1 i + a_2 j+ a_3 k is \sqrt{a_1^2+a_2^2+a_3^2}. This is sometimes called the modulus of the vector.

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