Exercise 29.1
Question 1
[निम्नलिखित प्रत्येक रेखा युग्म के बीच का कोण ज्ञात करें]
Find the angle between each of the following pair of lines :
(i)
\vec{r}=5 \hat{i}-7 \hat{j}+\lambda(-\hat{i}+4 \hat{j}+2 \hat{k})
\vec{r}=-2 \hat{i}+k+\mu(3 \hat{i}+4 \hat{k})
Sol :
(ii)
\vec{r}=(2+s) \hat{i}+(s-1) \hat{j}+(2-3 s) \hat{k}
\vec{r}=(1-t) \hat{i}+(2 t+3) \hat{j}+\hat{k}
Sol :
Question 2
[निम्नलिखित रेखा युग्मों के बीच का कोण ज्ञात कीजिए :]
Find the angle between the following pairs of lines :
(i)
\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-4}{6} तथा (and)
x+1=\frac{y+2}{2}=\frac{z-4}{2}
Sol :
(ii)
\frac{x+1}{3}=\frac{y-1}{5}=\frac{z+3}{4} तथा (and)
\frac{x+1}{1}=\frac{y-4}{4}=\frac{z-5}{2}
Sol :
Question 3
[निम्नलिखित रेखा युग्मों के बीच का कोण ज्ञात कीजिए ।]
Find the angle between the lines
(i) \frac{x+4}{3}=\frac{y-1}{5}=\frac{z+3}{4} and \frac{x+1}{1}=\frac{y-4}{1}=\frac{z-5}{2}
(ii) \frac{x-3}{1}=\frac{y-2}{2}=\frac{z-2}{-4} and \frac{x-0}{3}=\frac{y-5}{2}=\frac{z+2}{-6}
(iii) \vec{r}=4 \hat{i}-\hat{j}+\lambda(\hat{i}+2 \hat{j}-2 \hat{k}) and \vec{r}=\hat{i}-\hat{j}+2 \hat{k}-\mu(2 \hat{i}+4 \hat{j}-4 \hat{k})
Sol :
Question 4
Show that the line \frac{x-3}{2}=\frac{y+1}{-3}=\frac{z-2}{4} is perpendicular to the line \frac{x+2}{2}=\frac{y-4}{4}=\frac{z+5}{2}
Sol :
TYPE-II
Question 5
(i) If cartesian equation of a line is \frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}, find its vector equation.
(ii) If the cartesian equations of a line are \frac{3-x}{5}=\frac{y+4}{7}=\frac{2 z-6}{4}, write the vector equation for the line.
Sol :
Question 6
Find the equation of the line passing through point 2 \hat{i}-\hat{j}+4 \hat{k} and parallel to vector i+\hat{j}-2 \hat{k} in vector form as well as cartesian form.
Question 7
Find the equation of the line passing through points \hat{i}-2 \hat{j}+\hat{k} and -2 \hat{j}+3 \hat{k} in vector form and cartesian form.
Question 8
Find the equation of the line passing through point (1,0,2) having direction ratios 3,-1,5. Prove that this line passes through (4,-1,7)
Question 9
Find the equation of the line parallel to the line \frac{x-2}{3}=\frac{y+1}{1}=\frac{z-7}{9} and passing through the point (3,0,5)
Sol :
Question 10
Find the vector equation of a line passing through a point with position vector 2 \hat{i}-\hat{j}+\hat{k} and parallel to the line joining the points with position vectors -\hat{i}+4 \hat{j}+\hat{k} and \hat{i}+2 \hat{j}+2 \hat{k}. Also find the cartesian equation of this line.
Sol :
Question 11
A line passes through (2,-1,3) and is perpendicular to the lines \vec{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda(2 \hat{i}-2 \hat{j}+\hat{k}) and \vec{r}=(2 \hat{i}-\hat{j}-3 \hat{k})+\mu(\hat{i}+2 \hat{j}+2 \hat{k}). Obtain its equation in vector and cartesian form.
Sol :
Question 12
Find the vector equation of a line parallel to the vector 2 \hat{i}-\hat{j}+2 \hat{k} and passing through a point A with position vector 3 \hat{i}+\hat{j}-k.
Sol :
Question 13
Find the vector and cartesian equations of the line passing through the point (2,1,3) and perpendicular to the lines \frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3} and \frac{x}{-3}=\frac{y}{2}=\frac{z}{5}
Sol :
Question 14
The Cartesian equations of a line are :
\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}
Find a vector equation for the line.
Sol :
Question 15
Find the vector equation of a straight line which passes through the points whose position vectors are \hat{i}-2 \hat{j}+\hat{k} and 3 \hat{k}-2 \hat{j}.
Question 16
Find the vector equation of the straight line passing through the points :
(i) (1,1,0) and (0,1,1)
(ii) (-2,1,3) and (3,1,-2)
(iii) Find the cartesian equation of the line which passes through the point (-2,4,-5) and is parallel to the line \frac{x+3}{3}=\frac{4-y}{5}=\frac{z+8}{6}.
Sol :
TYPE-III
Question 17
(i) Find the point on the line \frac{x+2}{3}=\frac{y+1}{2}=\frac{z-3}{2} at a distance 3 \sqrt{2} from the point (1,2,3)
Sol :
(ii) Find the distance of the point (1,-2,3) from the plane x-y+z=5 measured parallel to the line \frac{x}{2}=\frac{y}{3}=\frac{z}{-6}.
Sol :
Question 18
Find the coordinates of the foot of perpendicular drawn from the point A(1,8,4) to the line joining B(0,-1,3) and C(2,-3,-1)
Sol :
Question 19
Find the image of the point (1,6,3) in the line \frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}.
Sol :
Question 20
Find the perpendicular distance of the point (1,0,0) from the line \frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+10}{8}.
Also find the coordinates of the foot of the perpendicular.
Sol :
Question 21
Find the foot of perpendicular from (0,2,7) to line
\frac{x+2}{-1}=\frac{y-1}{3}=\frac{z-3}{-2}
Sol :
Question 22
(i) Find the foot and hence the length of perpendicular from (5,7,3) to the line \frac{x-15}{3}=\frac{y-29}{8}=\frac{z-5}{-5}. Find also the equation of the perpendicular.
Sol :
(ii) Find the equation of the perpendicular drawn from (2,4,-1) to the line \frac{x+5}{1}=\frac{y+3}{4}=\frac{z-6}{-9}
Sol :
Question 23
Find two points on the line through the points A(1,2,3) and B(3,5,9) at a distance of 14 units from the mid point of AB
Sol :
Question 24
(i) Find the length of the perpendicular from point (3,4,5) on the line \frac{x-2}{2}=\frac{y-3}{5}=\frac{z-1}{3}.
Sol :
(ii) Find the length and the foot of the perpendicular drawn from the point (2,-1,5) to the line \frac{x-11}{10}=\frac{y+2}{-4}=\frac{z+8}{-11}
Sol :
TYPE-IV
Question 25
Find the shortest distance between the lines \frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4} and \frac{x-2}{3}=\frac{y-4}{4}=\frac{z-5}{5}. Find also its equation.
Sol :
Question 26
Find the shortest distance and equation of shortest distance between the lines \vec{r}=3 \hat{i}+5 \hat{j}+7 \hat{k}+\lambda(\hat{i}-2 \hat{j}+\hat{k}) and \vec{r}=-\hat{i}-\hat{j}-\hat{k}+\mu(2 \hat{i}-6 \hat{j}+\hat{k})
Sol :
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