Mathematics : Applications of Vector Algebra: Summary

**SUMMARY**

1. For a given set of three vectors , and , the scalar ( Ã— ). 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 |( Ã— ). |.

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 âˆˆ **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 Ã—( Ã— ) is called **vector triple product .**

7. For any three vectors **,** **,** we have Ã—( Ã— ) = (. ) - ( . ).

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 âˆˆ **R**.

9. Cartesian equations of a straight
line that passes through the point ( x_{1} , y_{1} , z_{1}
) and parallel to a vector with direction ratios b_{1} , b_{2}
, b_{3} are .

10. Any point on the line is
of the form ( x_{1} + tb_{1} , y_{1} + tb_{2} ,
z_{1} + tb_{3} ) , t âˆˆ
**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 ( x_{1} , y_{1} , z_{1}
) and ( x_{2} , y_{2} , z_{2} ) are

13. If Î¸ 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 (- ).( Ã— d ) = 0

19. The shortest distance between the two parallel lines where | | â‰ 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 ( x_{1}
, y_{1} , z_{1} ) is a ( x - x_{1} ) + b ( y - y_{1}
) + c ( z - z_{1} ) = 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 ( - ).( Ã— )
= 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 ( - ).( Ã— ) = 0 or ( - ).( Ã— ) = 0.

33. The acute angle Î¸ between the two
planes **.**

34. If Î¸ 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 (x_{2} , y_{1}, z_{1} ) 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 + d_{1} = 0 and ax + by + cz + d_{2} = 0 is
given by

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

, where Î» âˆˆ **R**
is an.

40. The equation of a plane passing
through the line of intersection of the planes *a _{1} x + b_{1} y + c_{1} z = d_{1} *and

*(a _{1}
x + b_{1} y + c_{1} z - d_{1} ) + Î»(a_{2} x + b_{2}
y + c_{2} z - d_{2} ) = 0*

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

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

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