Question

Find the (a)$\mathit{x}$, (b)$\mathit{y}$, and (c)$\mathit{z}$components of the sum of the displacements and whose components in meters are ${\mathit{c}}_{\mathbf{x}}\mathbf{=}\mathbf{7}\mathbf{.}\mathbf{4}\mathbf{,}\mathbf{}{\mathit{c}}_{\mathbf{y}}\mathbf{=}\mathbf{‐}\mathbf{3}\mathbf{.}\mathbf{8}\mathbf{,}\mathbf{}{\mathit{c}}_{\mathbf{z}}\mathbf{=}\mathbf{‐}\mathbf{6}\mathbf{.}\mathbf{1}\mathbf{;}\mathbf{}{\mathit{d}}_{\mathbf{x}}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{4}\mathbf{}\mathbf{,}\mathbf{}{\mathit{d}}_{\mathbf{y}}\mathbf{=}\mathbf{‐}\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{,}\mathbf{}{\mathit{d}}_{\mathbf{z}}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{3}$

Short answer

a) x component of is$c+d\mathrm{is}12\mathrm{m}$

b) y component of is$c+d\mathrm{is}‐5.8\mathrm{m}$

c) z component of is$c+d\mathrm{is}‐2.8\mathrm{m}$

Step-by-step solution

To calculate x component in meters

c and d are the displacement vectors. The sum of these vectors gives a resultant. Here the components of the resultant are to be found.

Given are the x, y z, components of the displacements c and d respectively.

$$\begin{gathered} c_x=7.4, \\ {c}_y=-3.8, \\ {c}_z=-6.1, \\ {d}_x=4.4, \\ {d}_y=-2.0, \\ {d}_y=3.3, \end{gathered}$$

The resultant of sum of displacements is given by

$$\begin{gathered} R=c+d \\ {R}_x=c_x+d_x \\ \left(\mathrm{i}\right) \\ {R}_y=c_y+d_y \\ \left(\mathrm{ii}\right) \\ {R}_z=c_z+d_z \\ \left(\mathrm{iii}\right) \end{gathered}$$

Substituting the above components in equation (i), the x component of R can be written as

$$\begin{gathered} R_x=7.4+4.4 \\ {R}_x=12\mathrm{m} \end{gathered}$$

To calculate y component in meters

$$\begin{gathered} R_y=-3.8-2.0 \\ {R}_y=-5.8\mathrm{m} \end{gathered}$$

To calculate z component in meters

$$\begin{gathered} R_z=-6.1+3.3 \\ {R}_z=-2.8\mathrm{m} \end{gathered}$$