Let _1 and _2 be the angles made by A and - A with the positive X… - Vidyadip

Question

Let $\varepsilon_1$ and $\varepsilon_2$ be the angles made by $\overrightarrow{\text{A}}$ and $-\overrightarrow{\text{A}}$ with the positive X-axis. Show that $\tan\varepsilon_1=\tan\varepsilon_2.$ Thus, giving tane does not uniquely determine the direction of $\overrightarrow{\text{A}}.$

✓

Answer

If a vector A makes α with x-axis then -A makes an angle $(\pi+\alpha)=\beta$ with same positive direction of x-axis.$\Rightarrow\tan\beta=\tan(\pi+\alpha)=\tan\alpha \ ...\text{QED}$
So yes if we only give $\tan\theta$ of a vector where $\theta$ is the angle made by the x-axis, it doesn’t uniquely determine the direction of a vector.

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