Question

If a body placed at the origin is acted upon by a force $\overline{F}=(\hat{i}+\hat{j}+\sqrt{2}\hat{k})$, then which of the following statements are correct?
1.Magnitude of $\overline{F}$ is $(2+sqrt2)$
2.Magnitude of $\overline{F}$ is 2
3. $\overline{F}$ makes an angle of ${45}^0$ with the Z-axis.
4. $\overline{F}$ makes an angle of ${30}^0$ with the Z-axis.
Select the correct answer using the codes given below.

• A. 1 and 3
• B. 2 and 3
• C. 1 and 4
• D. 2 and 4

Answer 1

The correct option is B 2 and 3

$\overrightarrow{F}=\hat{i}+\hat{j}+\sqrt{2}\hat{k}$

$|\overrightarrow{F}|=\sqrt{(1{)}^2+(1{)}^2+(\sqrt{2}{)}^2}$

$=\sqrt{1+1+2}$

$=\sqrt{4}=2\Rightarrow 2$

Now $\hat{F}=\frac{\overrightarrow{F}}{|\overrightarrow{F}|}=\frac{\hat{i}+\hat{j}+\sqrt{2}\hat{k}}{2}=\frac{1}{2}\hat{i}+\frac{1}{2}\hat{j}+\frac{\sqrt{2}}{2}\hat{k}...\left(1\right)$

Let $\alpha ,\beta ,\gamma$ be angles by $F$ from $x,y$ and $z$ axis respectively

So, $\hat{F}=\cos \alpha \hat{i}+\cos \beta \hat{j}+\cos \gamma \hat{k}....\left(2\right)$

From $\left(1\right)$ and $\left(2\right)$

$\cos \alpha =\frac{1}{2},\cos \beta =\frac{1}{2}$ and $\cos \gamma =\frac{1}{\sqrt{2}}$

$\Rightarrow \gamma ={45}^o\Rightarrow 3$

Ans $\left(B\right)2$ and $3$