Inter 1st Year Maths 1A Addition of Vectors Formulas

Use these Inter 1st Year Maths 1A Formulas PDF Chapter 4 Addition of Vectors to solve questions creatively.

Intermediate 1st Year Maths 1A Addition of Vectors Formulas

Scalar :
A physical quantity which has only magnitude is called a scalar quantity. All the real numbers will be taken as scalars.

Vector :
A physical quantity which has both magnitude and direction.
e.g. : Velocity acceleration, force, momentum.

Position vector :
Let ‘O’ and ‘P be any points in space. Then OP is called the position vector of the point ‘P w.r.t. origin ‘O’.
Note : \(\overline{A B}\) = Position vector of B- position vector of A’
= \(\overline{O B}-\overline{O A}\)

Coinitial vector:
Vectors having the same initial point are called coinitial vectors.
e.g. : \overline{O A}, \overline{O B}, \overline{O C} etc.

Unit vector:
A vector whose magnitude is one-unit is called unit vector.
Unit vector in the direction of a is denoted by â = \(\frac{\bar{a}}{|a|}\).
For any non-zero vector \(\bar{a}=|\bar{a}|\) â.

Inter 1st Year Maths 1A Addition of Vectors Formulas

Like vectors :
If two vectors are parallel and having the same direction then they are called like vectors.

Unlike vectors :
If two vectors are parallel and having Opposite direction then they are called unlike vectors.
The position vector of any point C on \(\overline{A B}\) can be taken as λ\(\bar{a}\) + µ\(\bar{a}\), where λ + µ = 1

Angle between vectors:
Let \(\overline{O A}=\bar{a}\) = a, \(\overline{O B}=\bar{a}\) = b be any two non – zero vectors, then angle AOB is defined as angle between vectors \(\bar{a}, \bar{b}\) and is denoted by \((\bar{a}, \bar{b})\) where 0 ≤ \((\bar{a}, \bar{b})\) ≤ 180°.

Addition of vectors (or) Parallelogram Law:
If \(\bar{a}, \bar{b}\) are the adjacent sides of a parallelogram the diagonals which is coinitial with \(\bar{a}, \bar{b}\) is given by \(\bar{a}+ \bar{b}\) and its magnitude is given by
\(|\bar{a}+\bar{b}|=\sqrt{|\bar{a}|^{2}+|\bar{b}|^{2}+2|\bar{a}||\bar{b}| \cos (\bar{a}, \bar{b})}\)

Triangle law:
If two vectors are represented in magnitude and direction by the two sides of a triangle taken in the same order, then their sum is represented by the third side taken in the reverse order.

Note : Addition of vectors is of 2 types :

  1. Commutative
  2. Associative.

(ie) (i) \(\bar{a}+\bar{b}=\bar{b}+\bar{a}\) and (ii) \((\bar{a}+\bar{b})+\bar{c}=\bar{a}+(\bar{b}+\bar{c})\)
Rule: \(|\bar{a}| \sim|\bar{b}| \leq|\bar{a}-\bar{b}| \leq|\bar{a}+\bar{b}| \leq|\bar{a}|+|\bar{b}|\)

Parallel (or) Collinear vector:

  1. Two vectors a, b are parallel or collinear, then a = λ. b, where ‘λ’ is a scalar.
  2. If \(\bar{a}\) = (a1, a2, a3), b = (b1, b2, b3) are parallel or collinear, then
    \(\frac{a_{1}}{b_{1}}=\frac{a_{2}}{b_{2}}=\frac{a_{3}}{b_{3}}\)
    Note : Zero vector is parallel to any vector.
  3. If three points with position vectors a, b, c are to be collinear. The necessary and sufficient condition is that there exists scalars \(\bar{a}, \bar{b}, \bar{c}\) not all zero, such that
    l\(\bar{a}\) + m\(\bar{b}\) + n\(\bar{c}\) = 0, l + m + n = 0.
  4. If \(\bar{a}, \bar{b}, \bar{c}\) are non-zero, non-collinear vectors such that l\(\bar{a}\) + m\(\bar{b}\) + n\(\bar{c}\) = 0, then l = 0, m = 0, n = 0.

Linear combination of vectors :
A linear combination of the system of vectors \(\bar{a}_{1}, \bar{a}_{2}, \ldots, \bar{a}_{n}\) is a vector.
r = x1\(\bar{a}_{1}\) + x2\(\bar{a}_{2}\) + x3\(\bar{a}_{3}\) + ………………. + xn\(\bar{a}_{n}\)
where x1 x2, x3, ………………., xn are scalars.

Coplanar vectors:
If three or more vectors lie in the same plane (or) parallel to the same plane then they are called coplanar vectors. If one vector can be expressed as a linear combination of the remaining vectors, then the vectors are coplanar vectors.
If \(\bar{a}\) = x\(\bar{b}\) + y\(\bar{c}\), where x, y, are scalars, then \(\bar{a}, \bar{b}, \bar{c}\) are coplanar.

Inter 1st Year Maths 1A Addition of Vectors Formulas

Linearly dependent system of vectors :
A system of vectors \(\bar{a}_{1}, \bar{a}_{2}, \bar{a}_{3}, \ldots \ldots, \bar{a}_{n}\) is said to
be linearly dependent, if there exists a system of scalars x1, x2, x3, ……………….., xn not all zero
such that
x1\(\bar{a}_{1}\) + x2\(\bar{a}_{2}\) + x3\(\bar{a}_{3}\) + ………………. + xn\(\bar{a}_{n}\) = \(\bar{0}\)

  • The null vector is linearly dependent.
  • Two collinear vectors are linearly dependent.
    \(\bar{a}\) = λ\(\bar{b}\) ⇒ (1)\(\bar{a}\) + (-λ)\(\bar{b}\) = 0
  • Any three coplanar vectors are linearly dependent.
    \(\bar{a}\) = x\(\bar{b}\) + y\(\bar{c}\) ⇒ (1)\(\bar{a}\) + (-x)\(\bar{b}\) + (-y)\(\bar{c}\) = 0
  • Any four vectors in space from a linearly dependent set of vectors.

Linearly independent system of vectors:
A system of vectors \(\bar{a}_{1}, \bar{a}_{2}, \bar{a}_{3}, \ldots \ldots, \bar{a}_{n}\) is said to be linearly independent, if x1\(\bar{a}_{1}\) + x2\(\bar{a}_{2}\) + x3\(\bar{a}_{3}\) + ………………. + xn\(\bar{a}_{n}\) = \(\bar{0}\) implies

  • Two non-zero, non-collinear vectors are linearly independent.
  • Any three non-coplanar vectors are linearly independent. If a, b, c are three non-coplanar vectors and \(\bar{r}\) be any other vector. Then there exists unique scalars x, y, z such that
    \(\bar{r}\) = x\(\bar{a}\) + y\(\bar{b}\) + z\(\bar{c}\).

Right handed system of vectors:
Three non-coplanar vectors \(\bar{a}, \bar{b}, \bar{c}\) are said to form a – right-handed system, if the rotation is form \(\bar{a}\) to \(\bar{b}\) in anti-clockwise direction, through an angle less than 180° as seen from the terminal point of c.
If \(\bar{a}, \bar{b}, \bar{c}\) from RHS and \(\bar{b}, \overline{,}, \bar{a}\) and \(\bar{c}, \bar{a}, \bar{b}\) also from RHS.

Left-handed system of vectors:
Three non-coplanar vectors \(\bar{a}, \bar{b}, \bar{c}\) are said to form a left handed system. If the rotation is from \(\bar{a}\) to \(\bar{b}\) in clockwise direction through an angle less than 180° as seen from terminal point of c.

If \(\bar{a}, \bar{b}, \bar{c}\) are in RHS, then
\(\bar{b}, \overline{,}, \bar{a}\); \(\bar{c}, \bar{a}, \bar{b}\) also from R.H.S.
and \(-\bar{a}, \bar{b}, \bar{c} ; \bar{a},-\bar{b}, \bar{c} ; \bar{a}, \bar{b},-\bar{c}\) form L.H.S.

Direction cosines of a vector and direction ratios of a vector:
If α, β, γ are the angles made by a line with +ve directions of x, y, z axes respectively, then cos α, cos β, cos γ are known as the direction cosines of that line.
Generally the direction cosines are denoted by l, m, n and l2 + m2 + n2 = 1.
Any numbers proportional to the direction cosines are known as direction ratios.

→ Unit vector in the direction of \(\bar{a}\) = \(\frac{\bar{a}}{|\bar{a}|}\)

→ Unit vector parallel to the resultant of the vectors \(\frac{\bar{a}+\bar{b}+\bar{c}}{|\bar{a}+\bar{b}+\bar{c}|}\) is ± \(\frac{\bar{a}+\bar{b}+\bar{c}}{|\bar{a}+\bar{b}+\bar{c}|}\)

→ The vector parallel to the resultant of the vectors \(\frac{\bar{a}+\bar{b}+\bar{c}}{|\bar{a}+\bar{b}+\bar{c}|}\) and having magnitude λ is ±λ \(\frac{\bar{a}+\bar{b}+\bar{c}}{|\bar{a}+\bar{b}+\bar{c}|}\)

→ The position vector of the point which divides the line segment joining the points A and B whose position vectors are \(\bar{a}, \bar{b}\) respectively, internally in the ratio, l:m is \(\frac{m \overline{\mathrm{a}}+l \bar{b}}{l+m}\) externally in the ratio l: m is \(\frac{m \bar{a}-l \bar{b}}{m-l}\), m ≠ l.

→ The ratio in which the line joining the points A(x1 y1, z1) and B(x2, y2, z2) is divided by

  • xy – plane is – z1: z2
  • yz – plane is -x1 : x2
  • zx – plane is -y1: y2

Inter 1st Year Maths 1A Addition of Vectors Formulas

→ If ‘c’ is the mid-point of line AB, the position vector of ‘c’ is \(\overline{O C}=\frac{\overline{O A}+\overline{O B}}{2}\)
= \(\frac{\bar{a}+\bar{b}}{2}\)

→ If \(\bar{a}, \bar{b}\) are two unit vectors, then the unit vector along the bisector of the angle between \(\bar{a}, \bar{b}\) is given by
\(\bar{c}=\frac{\bar{a}+\bar{b}}{|\bar{a}+\bar{b}|}\)
or
\(\bar{c}=\frac{\bar{a}-\bar{b}}{|\bar{a}-\bar{b}|}\)

→ The Vector \(\bar{a}\) = (a1, a2, a3), \(\bar{b}\) = (b1, b2, b3), \(\bar{c}\) = (c1, c2, c3) are coplanar or linearly dependent iff \(\left|\begin{array}{lll}
a_{1} & a_{2} & a_{3} \\
b_{1} & b_{2} & b_{3} \\
c_{1} & c_{2} & c_{3}
\end{array}\right|\) = 0

→ The Vector \(\bar{a}\) = (a1, a2, a3), \(\bar{b}\) = (b1, b2, b3), \(\bar{c}\) = (c1, c2, c3) are non-coplanar or linearly independent iff \(\left|\begin{array}{lll}
a_{1} & a_{2} & a_{3} \\
b_{1} & b_{2} & b_{3} \\
c_{1} & c_{2} & c_{3}
\end{array}\right|\) ≠ 0

→ The necessary and sufficient condition for four points with position vectors \(\bar{a}, \bar{b}, \bar{c}, \bar{d}\) are coplanar is that there exists scalars l, m, n, p not all zero such that
l\(\bar{a}\) + m\(\bar{b}\) + n\(\bar{c}\) + p\(\bar{da}\) = \(\bar{0}\), l + m + n + p = 0.

→ If \(\bar{a}, \bar{b}, \bar{c}\) are three non-zero, non – coplanar vectors and x, y, z are three scalars such that x\(\bar{a}\) + y\(\bar{b}\) + z\(\bar{c}\) = 0, then x = 0, y = 0, z = 0.
Note : Collinearity implies coplanarity but coplanarity does not imply collinearity.

→ If \(\bar{a}\) = (a1, a2, a3), \(\bar{b}\) = (b1, b2, b3), \(\bar{c}\) = (c1, c2, c3) and if \(\left|\begin{array}{lll}
a_{1} & a_{2} & a_{3} \\
b_{1} & b_{2} & b_{3} \\
c_{1} & c_{2} & c_{3}
\end{array}\right|\) > 0 then \(\bar{a}, \bar{b}, \bar{c}\) are in R.H.S

→ If \(\bar{a}\) = (a1, a2, a3), \(\bar{b}\) = (b1, b2, b3), \(\bar{c}\) = (c1, c2, c3) and if \(\left|\begin{array}{lll}
a_{1} & a_{2} & a_{3} \\
b_{1} & b_{2} & b_{3} \\
c_{1} & c_{2} & c_{3}
\end{array}\right|\) < 0 then \(\bar{a}, \bar{b}, \bar{c}\) are in L.H.S

Inter 1st Year Maths 1A Addition of Vectors Formulas

→ If \(\bar{r}\) = x\(\bar{i}\) + y\(\bar{j}\) + z\(\bar{k}\), then |\(\bar{r}\)| = r = \(\sqrt{x^{2}+y^{2}+z^{2}}\) .

→ If l, m, n are the direction cosines of a line, then l2 + m2 + n2 = 1.

→ If a vector makes angles α, β, γ with co-ordinate axes, then

  • cos2 α + cos2 β + cos2 γ = 1
  • sin2 α + sin2 β + sin2 γ =2

→ The direction ratios of a vector \(\bar{r}\) = a\(\bar{i}\) + b\(\bar{j}\) + c\(\bar{k}\) are a, b, c.

→ If A = (x1, y1 z1), B = (x2, y2, z2) then the direction ratios of a vector
AB are x2 – x1, y2 – y1, z2 – z1

→ If a, b, c are the direction ratios of a vector then direction cosines of a vector are
\(\pm \frac{a}{\sqrt{a^{2}+b^{2}+c^{2}}}, \pm \frac{b}{\sqrt{a^{2}+b^{2}+c^{2}}}, \pm \frac{c}{\sqrt{a^{2}+b^{2}+c^{2}}}\)

→ If a, b, c are the direction ratios of a line then λa, λb, λc also become the direction ratios of that line where ‘λ’ is a non-zero scalar.

→ If \(\bar{r}\) = x\(\bar{i}\) + y\(\bar{j}\) + z\(\bar{k}\), then the direction cosines of a vector are \(\frac{a}{|\vec{r}|}, \frac{b}{|\vec{r}|}, \frac{c}{|\vec{r}|}\)

→ If \(\bar{r}\) = x\(\bar{i}\) + y\(\bar{j}\) + z\(\bar{k}\) makes angles α, γ with co-ordinate axes respectively, then

cos α = \(\frac{a}{|\vec{r}|}\),
cos β = \(\frac{b}{|\vec{r}|}\)
cos γ = \(\frac{c}{|\vec{r}|}\).

  • The direction cosines of x- axis are 1, 0, 0.
  • The direction cosines of y – axis are 0, 1,0.
  • The direction cosines of z- axis are 0, 0, 1.

→ If a vector make equal angles with co-ordinate axes then direction cosines of a vector are \(\pm \frac{1}{\sqrt{3}}, \pm \frac{1}{\sqrt{3}}, \pm \frac{1}{\sqrt{3}}\)

→ If l, m, n are direction cosines of a vector OP and ‘O’ is the origin OP = r, then P = (lr, mr, nr).

→ If 1, m, n are direction cosines of a vector, then the maximum value of lmn is \(\frac{1}{3 \sqrt{3}}\)

→ The maximum value of l + m + n = \(\frac{1}{3 \sqrt{3}}\)

→ The vector equation of a straight-line passing through the point whose position vector is a and parallel to the vector \(\bar{b}\) is \(\bar{r}=\bar{a}+t \bar{b}\), where ‘t’ is parameter.

→ In cartesian form its equation is \(\frac{x-a_{1}}{b_{1}}=\frac{y-a_{2}}{b_{2}}=\frac{z-a_{3}}{b_{3}}\)
where \(\bar{a}\) = (a1, a2, a3), \(\bar{b}\) = (b1, b2, b3).

Inter 1st Year Maths 1A Addition of Vectors Formulas

→ The vector equation of a straight-line passing through the point whose position vectors are
a and b is \(\bar{r}\) = (1 – t)\(\bar{a}\) (or) \(\bar{r}=\bar{a}+t(\bar{b}-\bar{a})\); where’t’ is a parameter.

→ In cartesian form \(\frac{x-a_{1}}{b_{1}-a_{1}}=\frac{y-a_{2}}{b_{2}-a_{2}}=\frac{z-a_{3}}{b_{3}-a_{3}}\)
where \(\bar{a}\) = (a1, a2, a3), \(\bar{b}\) = (b1, b2, b3).

→ Vector equation of the plane through the points whose position vectors are \(\bar{a}, \bar{b}, \bar{c}\) is \(\bar{r}=(1-s-t) \bar{a}+s \bar{b}+t \bar{c}\), where s and t are parameters (or)
\(\bar{r}=\bar{a}+s(\bar{b}-\bar{a})+t(\bar{c}-\bar{a})\).
where \(\bar{a}\) = (a1, a2, a3), \(\bar{b}\) = (b1, b2, b3), \(\bar{c}\) = (c1, c2, c3)

→ The vector equation of the plane through the point whose position vector is \(\bar{a}\) and parallel to the vectors \(\bar{b}\) and \(\bar{c}\) is \(\bar{r}=\bar{a}+s \bar{b}+t \bar{c}\), where ‘s’ and ‘t’ are parameter.
In cartesian form it is \(\left|\begin{array}{ccc}
x-a_{1} & y-a_{2} & z-a_{3} \\
b_{1} & b_{2} & b_{3} \\
c_{1} & c_{2} & c_{3}
\end{array}\right|\) = 0

→ Vector equation of the plane through the point’s whose position vectors are a, b and parallel to vector c is
\(\bar{r}=(1-s) \bar{a}+s \bar{b}+t \bar{c}\)
or
\(\bar{r}=\bar{a}+s(\bar{b}-\bar{a})+t \bar{c}\)
In cartesian form it is \(\left|\begin{array}{ccc}
x-a_{1} & y-a_{2} & z-a_{3} \\
b_{1}-a_{1} & b_{2}-a_{2} & b_{3}-a_{3} \\
c_{1} & c_{2} & c_{3}
\end{array}\right|\) = 0