Question

Vector$\overrightarrow{\mathbf{A}}$has x and y components of -8.70 cmand 15.0 cm, respectively; vector$\overrightarrow{\mathbf{B}}$has x and y components of 13.2 cm and -6.60 cm, respectively. If$\overrightarrow{\mathbf{A}}\mathbf{-}\overrightarrow{\mathbf{B}}\mathbf{+}\mathbf{3}\overrightarrow{\mathbf{C}}\mathbf{=}\mathbf{0}$, what are the components of$\overrightarrow{\mathbf{C}}$?

Short answer

The x component of the vector$\overrightarrow{\mathrm{C}}$is${\mathrm{C}}_{\mathrm{x}}=7.30\mathrm{cm}$

The y component of the vector $\overrightarrow{\mathrm{C}}$is${\mathrm{C}}_{\mathrm{y}}=-7.20\mathrm{cm}$

Step-by-step solution

Definition of vector

• A vector is a quantity with both magnitude and direction.
• It's usually represented by an arrow with the same direction as the amount and a length proportionate to the magnitude of the quantity.
• A vector does not have position, even though it has magnitude and direction.

Determining the vector forms

Let us consider,

xis the vector's x component $\overrightarrow{\mathrm{A}}$is${\mathrm{A}}_{\mathrm{x}}=-8.70\mathrm{cm}$

The vector's y component$\overrightarrow{\mathrm{A}}$ is${\mathrm{A}}_{\mathrm{y}}=15\mathrm{cm}$

x is the vector's x component $\overrightarrow{\mathrm{B}}$ is${\mathrm{B}}_{\mathrm{x}}=13.2\mathrm{cm}$

The vector's y component $\overrightarrow{\mathrm{B}}$ is${\mathrm{B}}_{\mathrm{y}}=-6.60\mathrm{cm}$

The provided two vectors have the following vector forms:

$\begin{array}{rcl}\overrightarrow{\mathrm{A}}& =& {\mathrm{A}}_{\mathrm{x}}\hat{\mathrm{i}}+{\mathrm{A}}_{\mathrm{y}}\hat{\mathrm{j}}\\ & =& (-8.7\mathrm{cm})\hat{\mathrm{i}}+(15.0\mathrm{cm})\hat{\mathrm{j}}\\ \overrightarrow{\mathrm{B}}& =& {\mathrm{B}}_{\mathrm{x}}\hat{\mathrm{i}}+{\mathrm{B}}_{\mathrm{y}}\hat{\mathrm{j}}\\ & =& (13.2\mathrm{cm})\hat{\mathrm{i}}+(6.60\mathrm{cm})\hat{\mathrm{j}}\end{array}$

Evaluating the x and y components

Using the equation provided,

$\begin{array}{rcl}\overrightarrow{\mathrm{A}}-\overrightarrow{\mathrm{B}}+3\overrightarrow{\mathrm{C}}& =& 0\\ 3\overrightarrow{\mathrm{C}}& =& \overrightarrow{\mathrm{B}}-\overrightarrow{\mathrm{A}}\\ \overrightarrow{\mathrm{C}}& =& \frac{1}{3}[\overrightarrow{\mathrm{B}}-\overrightarrow{\mathrm{A}}]\\ & =& \frac{1}{3}\left[\right(13.2\mathrm{cm}\hat{\mathrm{i}}-6.60\mathrm{cm}\hat{\mathrm{j}})-(-8.70\mathrm{cm}\hat{\mathrm{i}}+15.0\mathrm{cm}\hat{\mathrm{j}}\left)\right]\\ & =& \frac{1}{3}[21.9\mathrm{cm}\hat{\mathrm{i}}-21.60\mathrm{cm}\hat{\mathrm{j}}]\\ & =& 7.30\mathrm{cm}\hat{\mathrm{i}}-7.20\mathrm{cm}\hat{\mathrm{j}}\end{array}$

Thus,

The vector's x component$\overrightarrow{\mathrm{C}}$ is${\mathrm{C}}_{\mathrm{x}}=7.30\mathrm{cm}$

The vector's y component.$\overrightarrow{\mathrm{C}}$ Is${\mathrm{C}}_{\mathrm{y}}=-7.20\mathrm{cm}$