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# Two proper vectors $\bar a$ and $\bar b$ are connected by $\bar a + \bar b = - \bar b$ . The ratio of the magnitudes of the vectors $\bar a$ and $\bar b$ is (A) $1:1$ (B) $2:1$ (C) $1:2$ (D) $3:1$

Last updated date: 09th Aug 2024
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Hint The magnitude of a vector ignores the direction, thus, the sign of the vector. The ratio of two numbers A and B is given as ${\text{A:B}}$ . So we need to calculate the magnitude of one vector in terms of the other to get the ratio.

From the question, we have two proper vectors $\bar a$ and $\bar b$ to be connected by the expression
$\Rightarrow \bar a + \bar b = - \bar b$
To find the ratio of the magnitudes, we must first find the value or expression for the vector $\bar a$ .
Hence, calculating the vector $\bar a$ from the connection between the vectors above, we subtract the vector $\bar b$ from both sides of the equation. Thus we have that
$\Rightarrow \bar a = - \bar b - \bar b = - 2\bar b$
$\therefore \bar a = - 2\bar b$ . Obviously, this implies that the vector $\bar a$ is twice the negative of the vector $\bar b$ .
However, we are only to calculate the ratio of the magnitudes, hence we must first find the magnitude of the vector $\bar a$ .
The magnitude of the vector is given as
$\Rightarrow \left| {\bar a} \right| = 2\bar b$ . In calculating magnitudes, the direction of the vector is not considered. This reflects by ignoring the sign of the vector.
Now the ratio of the magnitudes of $\bar a$ and $\bar b$ can be given as
$\Rightarrow \left| {\bar a} \right|:\left| {\bar b} \right|$ . Since the magnitude of $\bar a$ is $\left| {\bar a} \right| = 2\bar b$ , then
$\Rightarrow \left| {\bar a} \right|:\left| {\bar b} \right| = 2:1$
We can cancel the vector $\bar b$ , hence,
$\Rightarrow \left| {\bar a} \right|:\left| {\bar b} \right| = 2:1$
Hence, the correct answer is option B.

Note
Alternatively, to obtain the magnitudes, we can also express vector $\bar b$ in terms of $\bar a$ as in:
$\bar a = - 2\bar b$
$\Rightarrow \bar b = - \dfrac{{\bar a}}{2}$ . Following the same path. By finding the magnitude of the vector $\bar b$ . Hence, we have that
$\left| {\bar b} \right| = \dfrac{{\bar a}}{2}$ . Thus finding the ratio of the magnitudes, we have that
$\left| {\bar a} \right|:\left| {\bar b} \right| = \bar a:\dfrac{{\bar a}}{2}$ . Similarly, by eliminating the vector, we have
$\left| {\bar a} \right|:\left| {\bar b} \right| = 1:\dfrac{1}{2}$ . Multiplying by 2 we have
$\left| {\bar a} \right|:\left| {\bar b} \right| = 2:1$ which is identical to the answer above.