#### Exercise 11.1

**Question 1. If a line makes angles 90º, 135º, 45º with the x, y and z axes respectively, findits direction cosines.**

**Question 2. Find the direction cosines of a line which makes equal angles with coordinate axes.**

**Question 3. If a line has the direction ratios – 18, 12, –4 then what are its direction cosines?**

**Question 4. Show that the points (2, 3, 4) (–1, –2, 1), (5, 8, 7) are collinear.**

**Note:** For three points A, B and C, if direction ratios of AB and BC are proportional, then the three points are collinear.

**Question 5. Find the direction cosines of the sides of the triangle whose vertices are(3, 5, –4), (–1, 1, 2) and (–5, –5, –2).**

#### Exercise 11.2

**Question 1. Show that the lines with direction cosines:**

**Question 2. Show that the line through the points (1, –1, 2) (3, 4, –2) is perpendicular tothe line through the points (0, 3, 2) and (3, 5, 6).**

**Question 3. Show that the line through the points (4, 7 , 8) (2, 3, 4) is parallel to the linethrough the points (–1, –2, 1) and (1, 2, 5).**

**Question 4. Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector 3i+ 2j -2k.**

**Question 5. Find the equation of the line in vector and in cartesian form that passesthrough the point with position vector 2i -j +4k and is in the direction i +2j -k.**

**Question 6. Find the cartesian equation of the line which passes through the point**

**Question 7. The cartesian equation of a line is x-5/3 = y+4/7 = z-6/2. write its vector form.**

**Question 8. Find the vector and the cartesian equations of the lines that passes throughthe origin and (5, –2, 3).**

**Question 9. Find the vector and cartesian equations of the line that passes through thepoints (3, –2, –5), (3, –2, 6).**

**Question 10. Find the angle between the following pair of lines**

**Question 11. Find the angle between the following pair of lines**

**Question 12. Find the values of p so that the lines 1-x/3 = 7y-14/2p = z-3/2 and**

**Note: **For a line to be in standard form, coefficients of variables should be positive and constant.

**Question 13. Show that the lines x-5/7 = y+2/-5 and x/1 = y/2 = z/3 are perpendicular toeach other.**

**Question 14. Find the shortest distance between the lines**

**Question 15. Find the shortest distance between the lines x+1/7 = y+1/-6 = z+1/1 and x-3/1 = y-5/-2 = z-7/1.**

**Note: **If the two lines are parallel and non-intercepting, then only we can find the shortest distance between them.

**Question 16. Find the distance between the lines whose vector equations are:**

**Question 17. Find the shortest distance between the lines whose vector equations are**

#### Exercise 11.3

**Question 1. In each of the following exercises, determine the direction cosines of thenormal to the plane and the distance from the origin.**

**Question 2. Find the vector equation of a plane which is at a distance of 7 units from theorigin and normal to the vector 3i+ 5j – 6k.**

**Question 3. Find the Cartesian equation of the Following planes**

**Question 4. In the following cases find the coordinates of the foot of perpendicular drawn from the origin**.

**Question 5. Find the vector and cartesian equation of the planes**

**Question 6. Find the equations of the planes that passes through three points**

**Question 7. Find the intercepts cut off by the plane 2x + y – z = 5.**

**Note: **Intercepts cut of by any plane are the points where the plane meets the coordinate axes.

**Question 8. Find the equation of the plane with intercept 3 on the y- axis and parallel to ZOX plane.**

**Sol.** Any plane parallel to ZOX– plane is y = b where b is the intercept on y–axis. :. b = 3.Hence equation of the required plane is y = 3

**Question 9. Find the equation of the plane through the intersection of the planes 3x – y +2z –4 = 0 and x + y + z –2 = 0 and the point (2, 2, 1).**

**Question 10. Find the vector equation of the plane passing through the intersection of the**

**Question 11. Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0.**

**Question 12. Find the angle between the planes whose vector equations are**

**Question 13. In the following determine whether the given planes are parallel orperpendicular, and in case they are neither, find the angle between them.**

**Question 14. In the following cases, find the distance of each of the given points from the corresponding given plane.**

#### Miscellaneous Exercise

**Question 1. Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, –1), (4, 3, –1).**

** Sol.** First two points A and B are (0, 0, 0) and (2, 1, 1) respectively.

:. direction ratios of AB are 2, 1, 1

Direction ratios of CD joining the points C (3, 5, –1), D (4, 3, –1) are 1, –2, 0

Now, a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2}= 2 × 1 + 1 × (–2) + 1× 0 = 2 – 2 + 0 = 0

:. AB is perpendicular to CD.

**Question 2. If l _{1}, m_{1}, n_{1} and l_{2}, m_{2}, n_{2} are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m_{1} n_{2} – m_{2} n_{1}, n_{1} l_{2} – n_{2} l_{1}, l_{1} m_{2} – l_{2} m_{1}**

**Question 3. Find the angle between the lines whose direction ratios are a, b, c and b – c, c – a, a – b.**

**Question 4. Find the equation of a line parallel to x-axis and passing through the origin.**

**Sol. **The line parallel to x-axis and passing through the origin is x-axis itself.

:. equation of the line is y = 0, z = 0

**Question 5. If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (–4, 3, –6) and(2, 9, 2) respectively, then find the angle betwen the lines AB and CD.**

**Sol.** Direction ratios of AB when A and B are (1, 2, 3), (4, 5, 7) are 4 – 1, 5 – 2, 7–3 or 3, 3, 4 Direction ratios of the line joining the points C (–4, 3, –6) and D (2, 9, 2)are 2 + 4, 9 – 3, 2 + 6 or 6, 6, 8

Direction ratio of AB and CD are proportional.

=> Angle between these lines is zero. Þ The lines AB and CD are parallel.

**Question 6. If the lines x-1/-3 = y-2/2k = z-3/2 and x-1/3k = y-1/1 = z-6/-5 are perpendicular, find the value of k.**

**Question 7. Find the vector equation of the line passing through (1, 2, 3) and perpendicular to the plane r (i +2j -5k) + 9 = 0**

**Question 8. Find the equation of the plane passing through (a, b, c) and parallel to the plane r (i +j +k) = 2.**

**Question 9. Find the shortest distance between lines**

**Question 10. Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1)crosses the YZ-plane.**

**Question 11. Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1)crosses the ZX – plane.**

**Question 12. Find the coordinates of the point where the line through (3, – 4, –5) and (2, –3, 1) crosses the plane 2x + y + z = 7.**

**Question 13. Find the equation of the plane passing through the point (–1, 3, 2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0.**

**Question 14. If the points (1, 1, p) and (–3, 0, 1) be equidistant from the planer r. (3i+ 4j -12k) + 13 = 0, then find the value of p.**

**Question 15. Find the equation of the plane passing through the line of intersection of the**

**Question 16. If O be the origin and the coordiantes of P be (1, 2, –3), then find the equation of the plane passing through P and perpendicular to OP.**

**Sol. **The points O and P are (0, 0, 0) and (1, 2, –3)

direction ratios of OP are 1, 2, –3

The plane passing through (x1, y1, z1) is: a (x – x1) + b (y – y1) + c (z – z1) = 0 where a, b, c are the direction ratios of normal.

Direction ratios of normal are 1, 2, –3 and the point P is (1, 2, –3)

Equation of the required plane is 1 (x – 1) + 2 (y – 2) –3 (z + 3) = 0 or x + 2y – 3z – 14 = 0

**Question 17. Find the equation of the plane which contains the line of intersection of the**

**Question 18. Find the distance of the point (–1, –5, –10) from the point of intersection of**

**Question 19. Find the vector equation of the line passing through (1, 2, 3) and parallel to**

**Question 20. Find the vector equation of the line passing through the point (1, 2, –4) and perpendicular to the two lines:**

**Question 21. Prove that if a plane has the intercepts a, b, c and is at a distance of p units**

**Choose the correct answer in Questions 22 and 23.**

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