Question

A body moves through a distance of 5 m along the length, 4m along the breadthand 3 m along the height of a rectangular room. Find the displacement of thebody.​

Answer 1

Let a coordinate axes system be introduced in the rectangular room, where,

• x axis is introduced along the length of the room.

• y axis is introduced along the breadth of the room.

• z axis is introduced along the height of the room.

Let the displacement of the body be s (in m), which can be resolved into the three horizontal components $\sf{s_x,\ s_y}$ and $\sf{s_z}$ along the axes.

$\displaystyle\longrightarrow\mathbf{s}=s_x\,\mathbf{\hat i}+s_y\,\mathbf{\hat j}+s_z\,\mathbf{\hat k}$

Here,

• $\sf{s_x=5\ m}$

• $\sf{s_y=4\ m}$

• $\sf{s_z=3\ m}$

Then,

$\displaystyle\longrightarrow\mathbf{s}=5\,\mathbf{\hat i}+4\,\mathbf{\hat j}+3\,\mathbf{\hat k}$

For a vector $\mathbf{P}=P_x\,\mathbf{\hat i}+P_y\,\mathbf{\hat j}+P_z\,\mathbf{\hat k},$ we know that,

$\displaystyle\longrightarrow\left|\mathbf{P}\right|=\sqrt{\left(P_x\right)^2+\left(P_y\right)^2+\left(P_z\right)^2}$

Hence the magnitude of the displacement is,

$\displaystyle\longrightarrow|\mathbf{s}|=\sqrt{5^2+4^2+3^2}$

$\displaystyle\longrightarrow|\mathbf{s}|=\sqrt{25+16+9}$

$\displaystyle\longrightarrow|\mathbf{s}|=\sqrt{50}$

$\displaystyle\longrightarrow\underline{\underline{|\mathbf{s}|=5\sqrt{2}\ m}}$