Question

Can the resultant of two vectors be zero ?
A. yes, when the two vectors are same in magnitude and direction
B. No
C. yes, when the two vectors are same in magnitude but opposite in sense
D. yes, when the two vectors are same in magnitude making an angle of ${\displaystyle \frac{2\pi }{3}}$ with each other

Answer 1

Hint: In order to solve this question, we should know that, a vector is a representation of a quantity having a magnitude as well as a direction in three dimensional space, we will first write the notion of a general vector and then figure out how two vectors resultant can produce a zero vector.

Complete step by step answer:
A general vector in three dimensional space is written in the form of $\overrightarrow{r}=a\hat{i}+b\hat{j}+c\hat{k}$ where a, b, c are the components along the three axes X, Y, Z respectively and I, j, k are the unit vectors along these three directions and magnitude of a vector is calculated as
$\left|\overrightarrow{r}\right|=\sqrt{a^2+b^2+c^2}$

Now, let us take a vector say $\overrightarrow{r}=a\hat{i}+b\hat{j}+c\hat{k}$ and also let us suppose the same vector in opposite direction and the vector say r’ can be written in opposite direction as
${\overrightarrow{r}}^{\prime }=-\overrightarrow{r}$
so this vector is
${\overrightarrow{r}}^{\prime }=-a\hat{i}-b\hat{j}-c\hat{k}$
Now the magnitude of vector r’ is,
$$\begin{gathered} \Rightarrow \left|{\overrightarrow{r}}^{\prime }\right|=\sqrt{a^2+b^2+c^2} \\ \Rightarrow \left|{\overrightarrow{r}}^{\prime }\right|=\left|\overrightarrow{r}\right| \end{gathered}$$

Hence, both vectors r and r’ have same magnitude but opposite in direction and their resultant say vector R can be written as
$\overrightarrow{R}=\overrightarrow{r}+{\overrightarrow{r}}^{\prime }$ and we have, ${\overrightarrow{r}}^{\prime }=-\overrightarrow{r}$
$\Rightarrow \overrightarrow{R}=\overrightarrow{r}-\overrightarrow{r}$
$\therefore \overrightarrow{R}=0$
So, two vector results can be zero if they have the same magnitude but opposite in direction.

Hence, the correct option is C.

Note: It should be remembered that, magnitude is the square of a number so while changing the direction of a vector the sign of components of a vector changes but still the square of a negative number is also positive that’s why the magnitude of two vector which are opposite in direction are always equal.