Vectors

Addition of Vectors and Multiplication of a Vector by a Scalar


  • The unit vector in the direction of \overrightarrow{a} is \hat{a} = \pm \frac{\overrightarrow{a}}{|\overrightarrow{a}|}
  • The unit vector parallel to resultant of the two vectors \overrightarrow{a} and \overrightarrow{b} is = \pm \left(\frac{\overrightarrow{a} + \overrightarrow{b}}{|\overrightarrow{a} + \overrightarrow{b}|}\right)
  • The unit vector in the XY-plane making an angle θ with the positive direction of x-axis is \overrightarrow{r} = \cos \theta \hat{i} + \sin \theta \hat{j}
  • The unit vector in the space making an angles α, β, γ with the positive direction of x, y, z-axis respectively is \overrightarrow{r} = \cos \alpha \hat{i} + \cos \beta \hat{j} + \cos \gamma \hat{k}
  • The unit vector which makes equal acute angles with the coordinate axes is \overrightarrow{r} = \frac{1}{\sqrt{3}}\hat{i} + \frac{1}{\sqrt{3}}\hat{j} + \frac{1}{\sqrt{3}}\hat{k}
  • If A vector \tt P\left(\overrightarrow{r}\right) which divides the line joining two points A(\overrightarrow{a}), B(\overrightarrow{b}) in the ratio m:n internally, then \overrightarrow{r} = \frac{m\overrightarrow{b} + n \overrightarrow{a}}{m + n}
  • If A vector \tt P\left(\overrightarrow{r}\right) which divides the line joining two points A(\overrightarrow{a}), B(\overrightarrow{b}) in the ratio m:n externally, then \overrightarrow{r} = \frac{m\overrightarrow{b} - n \overrightarrow{a}}{m - n}
  • The mid point of a line joining two points A (\overrightarrow{a}) \ {\tt and} \ B (\overrightarrow{b}) \ {\tt is} \ = \frac{\overrightarrow{a} + \overrightarrow{b}}{2}
  • The position vector of the centroid (G) of the triangle with vertices A (\overrightarrow{a}), \ B (\overrightarrow{b}), c (\overrightarrow{c}) is = \frac{\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}}{3}
  • The position vector of the centrid (G) of the tetrahedron with vertices A (\overrightarrow{a}), B (\overrightarrow{b}), C (\overrightarrow{c}), D (\overrightarrow{d}) is = \frac{\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} + \overrightarrow{d}}{4}
  • In the right angles triangle right angle at A orthocentre is A, circumcentre is mid point of hypotenuse
  • In the equilateral triangle orthocentre and circumcentre is equal to centroid (G)
  • Unit Vector bisecting the angle between the vectors \overrightarrow{a} \ {\tt and} \ \overrightarrow{b} \ {\tt is} \ \frac{\hat{a} + \hat{b}}{|\hat{a} + \hat{b}|}
    where \hat{a} = \frac{\overrightarrow{a}}{|\overrightarrow{a}|}, \hat{b} = \frac{\overrightarrow{b}}{|\overrightarrow{b}|}
  • The vector bisecting the angle between the two vectors \overrightarrow{a} and \overrightarrow{b} is = \lambda (\hat{a} + \hat{b})
    where \hat{a} = \frac{\overrightarrow{a}}{|\overrightarrow{a}|}, \hat{b} = \frac{\overrightarrow{b}}{|\overrightarrow{b}|}
  • If a vector \overrightarrow{a} = (a_{1}, a_{2}, a_{3}) is rotated about x-axis through an angle α, then the new coordinates of \overrightarrow{a} are
    (a1, a2 cos α + a3 sin α, − a2 sin α + a3 cos α)
  • If a Vector \overrightarrow{a} = (a_{1}, a_{2}, a_{3}) is rotated about y-axis through an angle α, then the new coordinates of \overrightarrow{a} are
    (−a3 sin α + a1 cos α, a2,  a3 cos α + a1 sin α)
  • If a Vector \overrightarrow{a} = (a_{1}, a_{2}, a_{3}) is rotated about z-axis through an angle α, then the new coordinates of \overrightarrow{a} are
    (a1 cos α + a2 sin α, − a1 sin α + a2 cos α, a3)
  • Let O be the origin, \overrightarrow{OA} = \overrightarrow{a}, \overrightarrow{OB} = \overrightarrow{b} be two vectors, then the point \overrightarrow{Oc} = p \overrightarrow{a} + q\overrightarrow{b} lies
    → inside ΔOAB, if p > 0, q > 0 and p + q < 1
    → outside ΔOAB, but inside \angle AOB if p > 0, q > 0 and p + q > 1
    → outside ΔOAB, but inside \angle OAB if p < 0, q > 0 and p + q < 1
    → outside ΔOAB, but inside \angle OBA if p > 0, q < 0 and p + q < 1
  • If \overrightarrow{a} and \overrightarrow{b} are the adjacent sides and \overrightarrow{d_{1}} and \overrightarrow{d_{2}} are the diagonals of a parallelogram then 

    \overrightarrow{d_{1}} = \overrightarrow{a} + \overrightarrow{b}
    \overrightarrow{d_{2}} = \overrightarrow{a} - \overrightarrow{b}
    \overrightarrow{a} = \frac{\overrightarrow{d_{1}} + \overrightarrow{d_{2}}}{2}
    \overrightarrow{b} = \frac{\overrightarrow{d_{1}} - \overrightarrow{d_{2}}}{2}

Part1: View the Topic in this video From 10:05 To 53:36

Part2: View the Topic in this video From 00:40 To 54:37

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1. Properties of Vector Addition:

    (i) a + b = b + a                                (commutativity)

    (ii) a + (b + c) = (a + b) + c               (associativity)

    (iii) a + 0 = a                                     (additive identity)

    (iv) a + (−a) = 0                                (additive inverse)

    (v) (k1 + k2)a = k1a + k2a                  (multiplication by scalars)    

    (vi) k(a + b) = ka + kb                       (multiplication by scalars)

    (vii) |a + b| ≤ |a| + |b| and |a − b| ≥ |a| − |b|

2. (i)  |λ a| = |λ||a|

    (ii)  λ0 = 0

    (iii)  m (−a) = − ma = −(ma)

    (iv)  (−m)(−a) = ma

    (v)  m(n a) = mn a = n(m a)

    (vi)  (m + n) a = m a + n a

    (vii)  m (a + b) = m a + m b