Download App

Three Dimensional Geometry — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsThree Dimensional Geometry5 Marks
Question
Find the shortest distance between the lines whose vector equations are:
$\vec{\text{r}}=\Big(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}\Big)+\lambda\Big(\hat{\text{i}}-3\hat{\text{j}}+2\hat{\text{k}}\Big)$
$\text{and}\ \vec{\text{r}}=4\hat{\text{i}}+5\hat{\text{j}}+6\hat{\text{k}}+\mu\Big(2\hat{\text{i}}+3\hat{\text{j}}+\hat{\text{k}}\Big)$
✓

Answer

Equation of the first line is $\vec{\text{r}}=\Big(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}\Big)+\lambda\Big(\hat{\text{i}}-3\hat{\text{j}}+2\hat{\text{k}}\Big)$
Comparing this equation with $\vec{\text{r}}=\vec{\text{a}_1}+\lambda\vec{\text{b}_1},$
$\vec{\text{a}_1}=\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}\ \text{and}\ \vec{\text{b}_1}=\hat{\text{i}}-3\hat{\text{j}}+2\hat{\text{k}}$
Again equation of second line $\vec{\text{r}}=\Big(4\hat{\text{i}}+5\hat{\text{j}}+6\hat{\text{k}}\Big)+\mu\Big(2\hat{\text{i}}+3\hat{\text{j}}+\hat{\text{k}}\Big)$
Comparing this equation with $\vec{\text{r}}=\vec{\text{a}_2}+\mu\vec{\text{b}_2},$
$\vec{\text{a}_2}=4\hat{\text{i}}+5\hat{\text{j}}+6\hat{\text{k}}\ \text{and}\ \vec{\text{b}_2}=2\hat{\text{i}}+3\hat{\text{j}}+\hat{\text{k}}$
Now shortest distance $(\text{d})=\frac{\Big|\Big(\vec{\text{a}_2}-\vec{\text{a}_1}\Big).\Big(\vec{\text{b}_1}\times\vec{\text{b}_2}\Big)\Big|}{\Big|\vec{\text{b}_1}\times\vec{\text{b}_2}\Big|}\ \ \ \ ...(\text{i})$
Here $\vec{\text{a}_2}-\vec{\text{a}_1}=\Big(4\hat{\text{i}}+5\hat{\text{j}}+6\hat{\text{k}}\Big)-\Big(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}\Big)$
$=3\hat{\text{i}}+3\hat{\text{j}}+3\hat{\text{k}}$
$\vec{\text{b}_1}\times\vec{\text{b}_2}=\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}}\\1&-3&2\\2&3&1\end{vmatrix}$
$=(-3-6)\hat{\text{i}}-(1-4)\hat{\text{j}}+(3+6)\hat{\text{k}}=-9\hat{\text{i}}+3\hat{\text{j}}+9\hat{\text{k}}$
$\Big|\vec{\text{b}}_1\times\vec{\text{b}}_2\Big|=\sqrt{(-9)^2+(3)^2+(9)^2}=\sqrt{171}=3\sqrt{19}$
$\Big(\vec{\text{a}_2}-\vec{\text{a}_1}\Big).\Big(\vec{\text{b}_1}\times\vec{\text{b}_2}\Big)=3\times(-9)+(3\times3)+(3\times9)$
$=-27+9+27=9$
Putting these values in eq.(i),
Shortest distance $(\text{d})=\frac{|9|}{3\sqrt{19}}=\frac{9}{3\sqrt{19}}=\frac{3}{\sqrt{19}}.$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Three Dimensional Geometry→Three Dimensional Geometry — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

The rate of increase of bacteria in a culture is proportional to the number of bacteria present and it is found that the number doubles in 6 hours. Prove that the bacteria becomes 8 times at the end of 18 hours.
→
Find the largest possible area of a right angled triangle whose hypotenuse is 5cm long.
→
Evaluate the following integrals:
$\int_{0}^\limits{\frac{\pi}{2}}\frac{\text{x}+\sin\text{x}}{1+\cos\text{x}}\text{ dx}$
→
Find the shortest distance between the lines whose vector equations are:
$\vec{\text{r}}=(1-\text{t})\hat{\text{i}}+(\text{t}-2)\hat{\text{j}}+(3-2\text{t})\hat{\text{k}}\ \text{and}$
$\vec{\text{r}}=(\text{s}+1)\hat{\text{i}}+(2\text{s}-1)\hat{\text{j}}-(2\text{s}+1)\hat{\text{k}}$
→
Evaluate the following integrals:
$\int\limits^{2\pi}_0\frac{\text{e}^{\sin\text{x}}}{\text{e}^{\sin\text{x}}+\text{e}^{-\sin\text{x}}}\text{ dx}$
→
Find the equation of a plane passing through the point $\text{P (6, 5, 9)}$ and parallel to the plane determined by the points $\text{A (3, –1, 2), B (5, 2, 4)}$ and $\text{ C (–1, –1, 6)}$ . Also find the distance of this plane from the point A.
→
Solve the following differential equation:
$2\text{xy dx}+(\text{x}^2+2\text{y}^2)\text{dy}=0$
→
Find the value of k if f(x) is continuous at $\text{x}=\frac{\pi}{2},$ where
$\text{f}\text{(x)}=\begin{cases}\frac{\text{k}\cos\text{x}}{\pi-2\text{x}}, &\text{ x}\neq\frac{\pi}{2}\\3, &\text{ x}=\frac{\pi}{2}\end{cases}$
→
Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius $r$ is $\frac{4\text{r}}{3}.$ Also show that the maximum volume of the cone is$\frac{8}{27}$of the volume of the sphere.
→
Check whether the $\ast$ operation defined on the set $ \text{A = R} \times \text{R as} $
$\text{(a, b)} \ast \text{(c, d)} = \text{(a + c, b + d)}$
is a binary operation or not, where R is the set of all real numbers. If it is a binary operation, is it commutative and associative too? Also find the identity element of $\ast$.
→