Modulus of a Vector

The modulus of a vector is a positive number which is the measure of the length of the line segment representing that vector. For example, the modulus of a vector $\overrightarrow{OP}$ is the length $OP$ units and is written as $|\overrightarrow{OP}|$ (read modulus of $\overrightarrow{OP}$).

Let $P(x,y)$ be a point in the xy plane. Then, \[\overrightarrow{OP}=(x,y)\]

From $P$, draw $PM\perp OX$. Now, $OM=x$ and $MP=y$. From the right angled $\Delta OMP$, \[OP^2=OM^2+MP^2=x^2+y^2\] \[\therefore OP=\sqrt{x^2+y^2}\]

Hence, if $\overrightarrow{OP}=(x,y)$, then its modulus denoted by $|\overrightarrow{OP}|$ or $OP$ is given by \[OP=|\overrightarrow{OP}|=\sqrt{x^2+y^2}\]

In the case of space vector, the distance of any point $P(x,y,z)$ from the origin $O(0,0,0)$ is given by \[OP^2=x^2+y^2+z^2\] Hence, \[OP=|\overrightarrow{OP}|=\sqrt{x^2+y^2+z^2}\]

Modulus of a Space Vector or a Line

Let $O$ be the origin and let $OX$, $OY$ and $OZ$ represent x-axis, y-axis and z-axis respectively. Let $P(x,y,z)$ be a point in the space, then \[OA=x,\;\;OB=y\;\;\text{and}\;\;OC=z\]

Let $\overrightarrow{r}=\overrightarrow{OP}=(x,y,z)$$=x\:\overrightarrow{i}+y\:\overrightarrow{j}+ z\:\overrightarrow{k}$ From the right angled triangle $OMP$, \[OP^2=OM^2+MP^2=OM^2+OC^2\] \[=OA^2+OB^2+OC^2\] \[=x^2+y^2+z^2\] \[\therefore OP=\sqrt{x^2+y^2+z^2}\]

Thus, if $\overrightarrow{OP}=\overrightarrow{r}=x\:\overrightarrow{i}+y\:\overrightarrow{j}+ z\:\overrightarrow{k}\;$, then the modulus of $\overrightarrow{OP}$ is given by, \[OP=|\overrightarrow{OP}| =\sqrt{x^2+y^2+z^2}\] i.e. \[OP=\sqrt{\text{(coeff. of }\overrightarrow{i}\text{)}^2+\text{(coeff. of }\overrightarrow{j}\text{)}^2+\text{(coeff. of }\overrightarrow{i}\text{)}^2}\]