find shortest distance between the line :

x - 8/ 3 = y + 19/ -16 = z - 10/ 7 and x - 15/ 3 = y - 29 /8 = z - 5 / -5

Question

find shortest distance between the line :
x - 8/ 3 = y + 19/ -16 = z - 10/ 7 and x - 15/ 3 = y - 29 /8 = z
- 5 / -5

Gursheen kaur. · Accepted Answer

The equations of two given lines are:

$\frac{x - 8}{3} = \frac{y + 19}{- 16} = \frac{z - 10}{7} .......... (i) & \frac{x - 15}{3} = \frac{y - 29}{8} = \frac{z - 5}{- 5} .......... (i i)$

Line (i) passes though (8,-19,10) and has direction ratios proportional to 3,-16,7.

So, its vector equation is

$\vec{r} = {\vec{a}}_{1} + λ {\vec{b}}_{1} ............. (i i i) w h e r e, {\vec{a}}_{1} = 8 \hat{i} - 19 \hat{j} + 10 \hat{k} & {\vec{b}}_{1} = 3 \hat{i} - 16 \hat{j} + 7 \hat{k}$

Line (ii) passes though (15,29,5) and has direction ratios proportional to 3,8,-5.

So, its vector equation is

$\vec{r} = {\vec{a}}_{2} + μ {\vec{b}}_{2} ............. (i v) w h e r e, {\vec{a}}_{2} = 15 \hat{i} + 29 \hat{j} + 5 \hat{k} & {\vec{b}}_{2} = 3 \hat{i} + 8 \hat{j} - 5 \hat{k}$

The shortest distance between the lines (iii) and (iv) is given by

$S . D . = | \frac{(\vec{a_{2}} - \vec{a_{1}}) . (\vec{b_{1}} \times \vec{b_{2}})}{| \vec{b_{1}} \times \vec{b_{2}} |} | N o w, \vec{a_{2}} - \vec{a_{1}} = 7 \hat{i} + 48 \hat{j} - 5 \hat{k}$

$\vec{b_{1}} \times \vec{b_{2}} = 24 \hat{i} + 36 \hat{j} + 72 \hat{k} | \vec{b_{1}} \times \vec{b_{2}} | = \sqrt{{(24)}^{2} + {(36)}^{2} + {(72)}^{2}} = 84 (\vec{a_{2}} - \vec{a_{1}}) . (\vec{b_{1}} \times \vec{b_{2}}) = (7 \hat{i} + 48 \hat{j} - 5 \hat{k}) . (24 \hat{i} + 36 \hat{j} + 72 \hat{k}) = 168 + 1728 - 360 = 1536 S . D . = \frac{1536}{84} = \frac{793}{42}$

$\vec{b_{1}} \times \vec{b_{2}} = 24 \hat{i} + 36 \hat{j} + 72 \hat{k} | \vec{b_{1}} \times \vec{b_{2}} | = \sqrt{{(24)}^{2} + {(36)}^{2} + {(72)}^{2}} = 84 (\vec{a_{2}} - \vec{a_{1}}) (\vec{b_{1}} \times \vec{b_{2}}) = (7 \hat{i} + 48 \hat{j} - 5 \hat{k}) (24 \hat{i} + 36 \hat{j} + 72 \hat{k}) = 168 + 1728 - 360 = 1536 S D = \frac{1536}{84} = \frac{793}{42}$

find shortest distance between the line : x - 8/ 3 = y + 19/ -16 = z - 10/ 7 and x - 15/ 3 = y - 29 /8 = z - 5 / -5

find shortest distance between the line :

x - 8/ 3 = y + 19/ -16 = z - 10/ 7 and x - 15/ 3 = y - 29 /8 = z - 5 / -5