Chapter 3, Problem 1Q

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?

To determine

To prove:

 The sum of the magnitudes of two vectors are be equal to the magnitude of the sum of the same two vectors.

$\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{c}}\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\mathrm{a}+\mathrm{b}=\mathrm{c}$

Answer to Problem 1Q

Solution:

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

Explanation of Solution

We can use formula of the addition of two vectors and find their magnitudes.

Formula:

$\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{c}}$

Calculations:

Consider $\overrightarrow{\mathrm{a}}=5\widehat{\mathrm{i}}$  and $\overrightarrow{\mathrm{b}}=4\widehat{\mathrm{i}}$  are acting along the same direction as x axis. The magnitudes are $\mathrm{a}=5$ and $\mathrm{b}=4$

The sum of the magnitude of two vectors:

$\mathrm{a}+\mathrm{b}=\mathrm{c}$

$5+4=9$

$\mathrm{c}=9\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\dots \left(1\right)$

The magnitude of the sum of two vectors:

According to the vector addition law,

$\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{c}}$

$5\widehat{\mathrm{i}}+4\widehat{\mathrm{i}}=\overrightarrow{\mathrm{c}}$

$9\widehat{\mathrm{i}}=\overrightarrow{\mathrm{c}}$

$\mathrm{c}=9\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\dots \left(2\right)$

Hence, two vectors are acting in the same direction, then $\mathrm{ }\mathrm{a}+\mathrm{b}=\mathrm{c}$ is proved.

Conclusion:

We can use 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 along the same direction.