Vector

Important Results:
1. Triangle law of vectors

If OA = a, AB = b then OB = OA + AB = a+b

2. If O is a fixed point and P any point, OP= r represents the position vector of P.

3. AB= position vector of B- position vector of A.

4. If P is (x, y, z) then P.V of P is x i + y j + z k

5. If r= x i + y j + z k then |r| = $\sqrt{x^{2} + y^{2} + z^{2}}$

6. a and b are collinear vectors, then a = mb where m is a scalar.

7. If a, b, c are coplanar, then any one of them can be expressed as a linear combination of the other two, ie, a= x b + y c.

8. To prove that A, B, C are collinear, find AB, BC and then prove that one of them is a scalar multiple of the other.

9. To prove that A, B, C, D are coplanar, find AB, AC, AD and then show that these are coplanar.

10. Section formula:

If a and b are P.V's of A and B, then P.V of a point dividing AB in the ratio I:m is

given bt r= $\frac{l b + m a}{l + m}$

11. P.V of mid point of AB = $\frac{a + b}{2}$

12. P.V of the centroid of a triangle ABC is $\frac{a + b + c}{3}$

13. Dot product or Scalar product:

a.b = |a| |b| cos $θ$

14. cos $θ$ = $\frac{a . b}{| a | | b |}$

15. Scalar projection of b in the direction of a = $\frac{a . b}{| a |}$

Note:
Components of a vector

a. Components of a vector r in the direction of 'a' is $[\frac{\bar{r} . \bar{a}}{{| \bar{a} |}^{2}}] a$

b. and perpendicular to a is $\bar{r}$ - $[\frac{\bar{r} . \bar{a}}{{| \bar{a} |}^{2}}] a$
16. a.b = b.a

17. If a and b are non zero vectors and a.b = 0 then a and b are perpendicular

18. $a^{2}$ = a. a= $a^{2}$

19. ${(a + b)}^{2}$ = $a^{2}$ + $b^{2}$ +2 a.b,

${(a - b)}^{2}$ = $a^{2}$ + $b^{2}$ -2 a.b and

$(a + b)$ .(a-b)= $a^{2}$ - $b^{2}$

20. a.(b+c)= a.b + a.c

21. If i, j, k are unit vectors along the x, y, z axes respectively, then

i.i = j.j = k.k = 1

i.j = j . i = j. k = k. j = k. i = i .k = 0

22. If a = $a_{1}$ i + $a_{2}$ j + $a_{3}$ k, b = $b_{1}$ i + $b_{2}$ j + $b_{3}$ k, then a. b = $a_{1} b_{1}$ + $a_{2} b_{2}$ + $a_{3} b_{3}$

23. cos $θ$ = $\frac{a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}}{\sqrt{Σ a_{1}^{2}} . \sqrt{Σ b_{1}^{2}}}$

24. Condition that a, perpendicular to b is $a_{1} b_{1}$ + $a_{2} b_{2}$ + $a_{3} b_{3}$ = 0

25. Cross Product (Vector Product)

a $\times$ b = |a| |b| sin $θ$ n, where n is a unit vector perpendicular to the plane of a nd b such

that a, b, n form a right handed triad.

26. a $\times$ b = vector area of the parallellogram whose adjacent sides are a and b.

27. $\frac{1}{2}$ (a $\times$ b) = vector area of the triangle whose adjacent sides are a and b.

28. Area of the parallelogram whose diagonals are $d_{1}$ and $d_{2}$ is $\frac{1}{2}$ | $d_{1}$ $\times$ $d_{2}$ |

29. If a= $a_{1}$ i + $a_{2}$ j + $a_{3}$ k and b = $b_{1}$ i + $b_{2}$ j + $b_{3}$ k, then

a $\times$ b= $| \begin{matrix} i & j & k \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \end{matrix} |$

30. a $\times$ b $\neq$ b $\times$ a and b $\times$ a = -a $\times$ b

31. a $\times$ a= 0

32. If a and b are collinear then a x b = 0

33. a $\times$ ( b +c ) = a $\times$ b + a $\times$ c

34. i $\times$ i= j $\times$ j = k $\times$ k= 0

35. i $\times$ j= k, j $\times$ k= i, k $\times$ i = j and j $\times$ i= -k, k $\times$ j= -i, i $\times$ k= -j

36. sin $θ$ = $\frac{| a \times b |}{| a | . | b |}$ unit vector $\bar{n}$ = $\frac{\bar{a} \times \bar{b}}{| \bar{a} \times \bar{b} |}$

37. Scalar Triple Product

(a $\times$ b) . c or a. ( b $\times$ c) is called the scalar triple product of a, b, c and is denoted by [a,

b, c] or [a b c]

38. [ a b c] = volume of the parallelopiped whose coterminous edges are a, b, c

39. [ a, b, c]= $| \begin{matrix} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{matrix} |$

40. [a, b, c] = [b, c, a] = [c, a, b]

41. [a, a, b] = [a, b, b] = [a, c, c] = 0

42. If a, b, c are non zero non parallel vectors then a, b, c are coplanar if [a, b, c] = 0

43. In scalar triple product, the dot and the cross can be interchanged ie, a.(b $\times$ c) = (a $\times$ b).c

44. Vector triple product

a $\times$ ( b $\times$ c) = (a.c) b- (a.b). c

45. (a $\times$ b). (c $\times$ d)= $| \begin{matrix} a . c & a . d \\ b . c & b . d \end{matrix} |$

46. Reciprocal System of vectors

If a, b, c and $a^{'}$ , $b^{'}$ , $c^{'}$ are reciprocal system of vectors where [a, b, c] $\neq$ 0

then $a^{'}$ = $\frac{b \times c}{[a, b, c]}$ , $b^{'}$ = $\frac{c \times a}{[a, b, c]}$ , $c^{'}$ = $\frac{a \times b}{[a, b, c]}$

47. If F is the force causing a displacement S, then work done = F.S

48. Moment of a force F about a point P is PQ x F where Q is a point on the line of action

of F.