31.

If A, B, C are three non- collinear points with position vectors $\overrightarrow{\mathrm{a}} , \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ respectively, then show that the length of the perpendicular from C on AB is $\frac{\left|(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) + (\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}) + (\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{a}}) \right|}{\left|\hat{\mathrm{b}} - \overrightarrow{\mathrm{a}}\right|}$

Show that the four points A,B, C and D with position vectors:

$4\hat{\mathrm{i}} + 5\hat{\mathrm{j}} + \hat{\mathrm{k}}, - \hat{\mathrm{j}} - \hat{\mathrm{k}}, 3\hat{\mathrm{i}} + 9\hat{\mathrm{j}} + 4\hat{\mathrm{k}} \mathrm{and} 4(-\hat{\mathrm{i}} +\hspace{0.17em}\hat{\mathrm{j}} +\hat{\mathrm{k}})$ respectively are coplanar.

Draw a rough sketch of the curve and find the area of the region bounded by curve y² = 8x and the line x =2.

Sketch the graph of y = |x + 4|. Using integration, find the area of the region bounded by the curve y = |x + 4| and x = – 6 and x = 0.

Find the image of a point having position vector: $3\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + \hat{\mathrm{k}}$ in the plane $\overrightarrow{\mathrm{r}}.\left(3\hat{\mathrm{i}} - \hat{\mathrm{j}} + 4\hat{\mathrm{k}}\right) = 2$.

Find the equation of the plane through the intersection of the planes $\overrightarrow{\mathrm{r}}.(\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - \hat{\mathrm{k}}) = 9 \mathrm{and} \overrightarrow{\mathrm{r}}.(2\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}) = 3$