Question

$$\begin{gathered} \mathbf{If}\mathbf{}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{d}}\mathbf{=}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{a}}\mathbf{+}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{b}}\mathbf{+}\left(-\stackrel{\rightharpoonup }{c}\right)\mathbf{}\mathbf{,}\mathbf{}\mathbf{does}\mathbf{}\left(a\right)\mathbf{}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{a}}\mathbf{+}\left(-\stackrel{\rightharpoonup }{d}\right)\mathbf{=}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{c}}\mathbf{+}\left(-\stackrel{\rightharpoonup }{b}\right)\mathbf{}\mathbf{,}\mathbf{}\left(b\right)\mathbf{}\mathbf{}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{a}}\mathbf{=}\left(-\stackrel{\rightharpoonup }{b}\right)\mathbf{+}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{d}}\mathbf{+}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{c}} \\ \stackrel{\mathbf{\rightharpoonup }}{\mathbf{c}}\mathbf{+}\left(-\stackrel{\rightharpoonup }{d}\right)\mathbf{=}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{a}}\mathbf{+}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{b}}\mathbf{?} \end{gathered}$$

Short answer

$$\begin{gathered} \left(\mathrm{a}\right)\stackrel{\rightharpoonup }{a}+\left(-\stackrel{\rightharpoonup }{d}\right)=\stackrel{\rightharpoonup }{c}+\left(-\stackrel{\rightharpoonup }{b}\right)\mathrm{is}\mathrm{valid} \\ \left(\mathrm{b}\right)\stackrel{\rightharpoonup }{\mathrm{a}}=\left(-\stackrel{\rightharpoonup }{\mathrm{b}}\right)+\stackrel{\rightharpoonup }{\mathrm{d}}+\stackrel{\rightharpoonup }{\mathrm{c}}\mathrm{is}\mathrm{valid} \\ \left(\mathrm{c}\right)\stackrel{\rightharpoonup }{\mathrm{c}}+\left(-\stackrel{\rightharpoonup }{\mathrm{d}}\right)=\stackrel{\rightharpoonup }{\mathrm{a}}+\stackrel{\rightharpoonup }{\mathrm{b}}\mathrm{is}\mathrm{not}\mathrm{valid} \end{gathered}$$

Step-by-step solution

Given information

The vector is given by,

$\stackrel{\rightharpoonup }{d}=\stackrel{\rightharpoonup }{a}+\stackrel{\rightharpoonup }{b}+\left(-\stackrel{\rightharpoonup }{c}\right)$

To understand the concept

The vector operations are different than the scalar operations. We cannot add or subtract the vectors like scalars. The vector operations are governed by the set of laws for vector addition and vector subtraction.

This problem involves a basic substation operation in which the given vector equation$\stackrel{\rightharpoonup }{d}=\stackrel{\rightharpoonup }{a}+\stackrel{\rightharpoonup }{b}+\left(-\stackrel{\rightharpoonup }{c}\right)$can be usedin each equation and checked on both sides of the equation.

Formula:

$\stackrel{\rightharpoonup }{a}+\stackrel{\rightharpoonup }{b}=\stackrel{\rightharpoonup }{c}$

(a) To check validity for a⇀+(-d⇀)=c⇀+(-b⇀)

$$\begin{gathered} \stackrel{\rightharpoonup }{a}+\left(\stackrel{\rightharpoonup }{a}-\stackrel{\rightharpoonup }{b}+\stackrel{\rightharpoonup }{c}\right)=\stackrel{\rightharpoonup }{c}+\left(-\stackrel{\rightharpoonup }{b}\right) \\ \left(-\stackrel{\rightharpoonup }{b}\right)+\stackrel{\rightharpoonup }{c}=\stackrel{\rightharpoonup }{c}+\left(-\stackrel{\rightharpoonup }{b}\right) \end{gathered}$$

As$\left(-\stackrel{\rightharpoonup }{b}\right)$andare commutative, so both sides are equal.

Hence, this is a correct equation.

(b) To check validity for a⇀=(-b⇀)+d⇀+c⇀

$$\begin{gathered} \stackrel{\rightharpoonup }{a}=\left(-\stackrel{\rightharpoonup }{b}\right)+\stackrel{\rightharpoonup }{a}+\stackrel{\rightharpoonup }{b}+\left(-\stackrel{\rightharpoonup }{c}\right)+\stackrel{\rightharpoonup }{c} \\ \stackrel{\rightharpoonup }{a}=\stackrel{\rightharpoonup }{a} \end{gathered}$$.

Here, both sides are equal.

So, this is a correct equation.

(c) To check validity for c⇀+(-d⇀)=a⇀+b⇀

$$\begin{gathered} \stackrel{\rightharpoonup }{c}+\left(-\stackrel{\rightharpoonup }{a}-\stackrel{\rightharpoonup }{b}+\stackrel{\rightharpoonup }{c}\right)=\stackrel{\rightharpoonup }{a}+\stackrel{\rightharpoonup }{b} \\ \left(-\stackrel{\rightharpoonup }{a}-\stackrel{\rightharpoonup }{b}\right)+2\stackrel{\rightharpoonup }{c}=\stackrel{\rightharpoonup }{a}+\stackrel{\rightharpoonup }{b} \end{gathered}$$

Here, both sides are not equal.

So, this is not a correct equation.