# 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.

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.

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

→ 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

→ 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).

→ 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