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Applications of Vector Algebra - Summary | 12th Mathematics : UNIT 6 : Applications of Vector Algebra

Chapter: 12th Mathematics : UNIT 6 : Applications of Vector Algebra

Summary

Mathematics : Applications of Vector Algebra: Summary

SUMMARY

1. For a given set of three vectors  and  , the scalar ( ×  ).  is called a scalar triple product of  , , .

2. The volume of the parallelepiped formed by using the three vectors , and  co-terminus edges is given by |( ×  ). |.

3. The scalar triple product of three non-zero vectors is zero if and only if the three vectors are coplanar.

4. Three vectors ,  are coplanar, if, and only if there exist scalars r, s, t R such that atleast one of them is non-zero and r + s + t.

5. If  and  are any two systems of three vectors, and if 


6. For a given set of three vectors  ,  , the vector ×( ×  ) is called vector triple product .

7. For any three vectors , ,  we have  ×( × )  = () - ( . ).

8. Parametric form of the vector equation of a straight line that passes through a given point with position vector  and parallel to a given vector  is  = t , where t R.

9. Cartesian equations of a straight line that passes through the point ( x1 , y1 , z1 ) and parallel to a vector with direction ratios b1 , b2 , b3 are .

10. Any point on the line  is of the form ( x1 + tb1 , y1 + tb2 , z1 + tb3 ) , t R.

11. Parametric form of vector equation of a straight line that passes through two given points with position vectors  and  is .

12. Cartesian equations of a line that passes through two given points ( x1 , y1 , z1 ) and ( x2 , y2 , z2 ) are 

13. If θ is the acute angle between two straight lines  =   + s and  =   + td , then 

14. Two lines are said to be coplanar if they lie in the same plane.

15. Two lines in space are called skew lines if they are not parallel and do not intersect

16. The shortest distance between the two skew lines is the length of the line segment perpendicular to both the skew lines.

17. The shortest distance between the two skew lines  is


18. Two straight lines  =  + s and  =  + t intersect each other if ( ).( × d ) = 0

19. The shortest distance between the two parallel lines  where | | ≠ 0

20. If two lines  intersect, then 

21. A straight line which is perpendicular to a plane is called a normal to the plane.

22. The equation of the plane at a distance p from the origin and perpendicular to the unit normal vector is  . = p ( normal form)

23. Cartesian equation of the plane in normal form is lx + my + nz = p

24. Vector form of the equation of a plane passing through a point with position vector  and perpendicular to  is ( - ).  = 0.

25. Cartesian equation of a plane normal to a vector with direction ratios a,b,c and passing through a given point ( x1 , y1 , z1 ) is a ( x - x1 ) + b ( y - y1 ) + c ( z - z1 ) = 0 .

26. Intercept form of the equation of the plane  = q , having intercepts a, b, c on the x, y, z axes respectively is .

27. Parametric form of vector equation of the plane passing through three given non-collinear points is 

28. Cartesian equation of the plane passing through three non-collinear points is


29. A straight will lie on a plane if every point on the line, lie in the plane and the normal to the plane is perpendicular to the line.

30. The two given non-parallel lines  =  + s and  =  + t are coplanar if ( -  ).( ×  ) = 0 .

31. Two lines  are coplanar if 

32. Non-parametric form of vector equation of the plane containing the two  coplanar lines  =  + t and  =  + t is ( -  ).( ×  ) = 0 or ( -  ).( ×  ) = 0.

33. The acute angle θ between the two planes .

34. If θ is the acute angle between the line  =  + t and the plane 

35. The perpendicular distance from a point with position vector  to the plane . = p is given by  

36. The perpendicular distance from a point (x2 , y1, z1 ) to the plane ax + by + cz = p is 

37. The perpendicular distance from the origin to the plane ax + by + cz + d = 0 is given by 

38. The distance between two parallel planes ax + by + cz + d1 = 0 and ax + by + cz + d2 = 0 is given by 

39. The vector equation of a plane which passes through the line of intersection of the planes 

, where λ R is an.

40. The equation of a plane passing through the line of intersection of the planes a1 x + b1 y + c1 z = d1 and a2 x + b2 y + c2 z = d2 is given by

(a1 x + b1 y + c1 z - d1 ) + λ(a2 x + b2 y + c2 z - d2 ) = 0

41. The position vector of the point of intersection of the line  =  + tb and the plane  =  = p is , where .  ≠  .

42. If  is the position vector of the image of  in the plane  = p ,then 


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