Question

Angle between 2 planes $\overline{r}.\widehat{n_1}=d_1,\overline{r}.\widehat{n_2}=d_2$, can always be given by

• A. $\theta =si{n}^{-1}|\widehat{n_1}\times \widehat{n_2}|$
• B. $\theta =co{s}^{-1}(\widehat{n_1}.\widehat{n_2})$
• C. $\theta =si{n}^{-1}|\widehat{n_1}.\widehat{n_2}|$
• D. $\theta =co{s}^{-1}|(\widehat{n_1}.\widehat{n_2})|$

Answer 1

The correct option is B $\theta =co{s}^{-1}(\widehat{n_1}.\widehat{n_2})$

We know the angle between the planes is same as the angle between their normals. Here the normals are $\widehat{n_1}$ and $\widehat{n_2}$. So we can get the angle between them by dot product,
$\overline{n_1}.\overline{n_2}=cos\theta$
$\Rightarrow \theta =co{s}^{-1}(\widehat{n_1}.\widehat{n_2})$
One Important thing to notice is that, we will lose information on angle by doing $si{n}^{-1}|\widehat{n_1}\times \widehat{n_2}|$ or $co{s}^{-1}|\widehat{n_1}.\widehat{n_2}|$ as it will restrict the value of angle to $(0,\frac{\pi }{2})$ in case 1 and $(\frac{-\pi }{2},\frac{\pi }{2})$ in case 2.