If $|\overline{\mathrm{u}}|=2$ and $\overline{\mathrm{u}}$ makes angles of $60^{\circ}$ and $120^{\circ}$…

If $|\overline{\mathrm{u}}|=2$ and $\overline{\mathrm{u}}$ makes angles of $60^{\circ}$ and $120^{\circ}$ with axes $\mathrm{OX}$ and OY in the origin, then $\bar{u}=$
  1. $\hat{\mathrm{i}}+\hat{\mathrm{j}}+\sqrt{2} \hat{\mathrm{k}}$
  2. $2(\hat{\mathrm{i}}+\hat{\mathrm{j}} \pm \sqrt{2} \hat{\mathrm{k}})$
  3. $2(\hat{\mathrm{i}}-\hat{\mathrm{j}}+\sqrt{2} \hat{\mathrm{k}})$
  4. $2(\hat{\mathrm{i}}-\hat{\mathrm{j}} \pm \sqrt{2} \hat{\mathrm{k}})$

Solution

We have $|\overline{\mathrm{u}}|=2$ and $\cos \alpha=60^{\circ}=\frac{1}{2}$ and $\cos \beta=\cos 120^{\circ}=-\frac{1}{2}$ Now $\cos ^2 \gamma=1-\left(\frac{1}{4}+\frac{1}{4}\right)=\frac{1}{2} \quad \Rightarrow \cos \gamma= \pm \frac{1}{\sqrt{2}}$ Thus direction of $\overline{\mathrm{u}}$ are $1,-1, \pm \sqrt{2}$ $\therefore \overline{\mathrm{u}}=2(\hat{\mathrm{i}}-\hat{\mathrm{j}} \pm \sqrt{2} \hat{\mathrm{k}})$

Asked in: MHT CET 2021 (22 Sep Shift 1)

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