Question

Can the sum of the magnitudes of two vectors ever be equal to the magnitude of the sum of the same two vectors? If no, why not? If yes, when?

Short answer

Yes, when two vectors are in the same direction, then the sum of the magnitudes of two vectors is equal to the magnitude of the sum of the same two vectors.

Step-by-step solution

Vector addition

The problem deals with the addition of two vectors. Two vectors can be added only when they are of the same type and nature. We can use the formula for the addition of the two vectors and their magnitudes.

Formula:

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

To find whether the sum of the magnitudes of two vectors ever be equal to the magnitude of the sum of the same two vectors

Consider $\stackrel{\rightharpoonup }{a}=5\hat{\mathrm{i}}\mathrm{and}\stackrel{\rightharpoonup }{\mathrm{b}}=4\hat{\mathrm{i}}$are acting along the same direction as x axis. The magnitudes are $\left|a\right|=5\mathrm{and}\left|b\right|=4$

The sum of the magnitude of two vectors

localid="1660881852509" $$\begin{gathered} \left|a\right|+\left|b\right|=\left|c\right| \\ 5+4=\left|c\right| \\ \left|c\right|=\left|\stackrel{\rightharpoonup }{a}\right|+\left|\stackrel{\rightharpoonup }{b}\right| \\ =9 \\ \left(\mathrm{i}\right) \end{gathered}$$

The magnitude of the sum of two vectors:

According to the vector addition law,

$$\begin{gathered} \stackrel{\rightharpoonup }{a}+\stackrel{\rightharpoonup }{b}=\stackrel{\rightharpoonup }{c} \\ 5\hat{\mathrm{i}}+4\hat{\mathrm{i}}=\stackrel{\rightharpoonup }{c} \\ 9\hat{i}=\stackrel{\rightharpoonup }{c} \\ \left|c\right|=\left|\stackrel{\rightharpoonup }{a}+\stackrel{\rightharpoonup }{b}\right| \\ =9 \end{gathered}$$

(ii)

Hence, if two vectors are acting in the same direction then$\left|\stackrel{\rightharpoonup }{a}\right|+\left|\stackrel{\rightharpoonup }{b}\right|=\left|\stackrel{\rightharpoonup }{a}+\stackrel{\rightharpoonup }{b}\right|$is proved.

We can use the expression of vector addition law and find their magnitudes. It indicates that the sum of the magnitudes of two vectors can be equal to the magnitude of the sum of the same two vectors when they are going in the same direction.