Question

Consider the position vector $OP$ of a point $P(x,y,z)$ as in figure shown below

If $P_1$ be the foot of perpendicular from the $P$ on the plane $XOY$, then which of the following options is not true?

• A. Component form of vector $OP($ or $r)=x\hat{i}+y\hat{j}+z\hat{k}$
• B. $x,y$ and $z$ are called scalar components of OP or (r)
• C. $x\hat{i},y\hat{j}$ and $z\hat{k}$ are called the vector components of $r$ along the respective axis
• D. The length of any vector $r=x\hat{i}+y\hat{j}+z\hat{k}$ is given by $|r|=\sqrt{x+y+z}$

Answer 1

Answer: (D)

Considering the position vector OP of a point $P(x,y,z)$ as in figure where $P_1$ be the foot of the perpendicular from $P$ on the plane $XOY$. We, thus, see that $P_{1}P$ is

parallel to $z$-axis. As $\hat{i},\hat{j}$ and $\hat{k}$ are the unit vectors along the $x,y$ and $z$-axes, respectively and by the definition of the coordinates of $P$, we have $P_{1}P=OR=z\hat{k}$.
Similarly, $Q{P}_1=OS=y\hat{j}$ and $OQ=x\hat{i}$.
Therefore, it follows that $O{P}_1=OQ+Q{P}_1=x\hat{i}+y\hat{j}$ and $OP=O{P}_1+P_{1}P=x\hat{i}+y\hat{j}+z\hat{k}$
Hence, the position vector of $P$ with reference to $O$ is given by
$OP(\text{ or }r)=x\hat{i}+y\hat{j}+z\hat{k}$
This form of any vector is called its component form. Here $x$, $y$ and $z$ are called as the scalar components of $r$ and $x\hat{i},y\hat{j}$ and $z\hat{k}$ are called the vector components of $r$ along the respective axes. Sometimes $x,y$ and $z$ are also termed as rectangular components.
The length of any vector $r=x\hat{i}+y\hat{j}+z\hat{k}$, is readily determined by applying the Pythagoras theorem twice. We note that in the right angle $△OQ{P}_1$ (fig.)
$\left|O{P}_1\right|=\sqrt{|OQ{|}^2+{\left|Q{P}_1\right|}^2}=\sqrt{x^2+y^2}$
and in the right angle $△O{P}_{1}P$. we have
$OP=\sqrt{{\left|O{P}_1\right|}^2+{\left|P_{1}P\right|}^2}=\sqrt{\left(x^2+y^2\right)+z^2}$
Hence, the length of any vector $r=x\hat{i}+y\hat{j}+z\hat{k}$ is given by
$|r|=|x\hat{i}+y\hat{j}+z\hat{k}|=\sqrt{x^2+y^2+z^2}$
So, only option (d) is incorrect.