Inter 1st Year Maths 1B Direction Cosines and Direction Ratios Formulas

   

Use these Inter 1st Year Maths 1B Formulas PDF Chapter 6 Direction Cosines and Direction Ratios to solve questions creatively.

Intermediate 1st Year Maths 1B Direction Cosines and Direction Ratios Formulas

Formulae and Synopsis :
If a line PQ makes angles α, β, γ with the co – ordinate axes, then cos α, cos β, cos γ are called
direction cosines of the line PQ. We take
l = cos α, m = cos β and n = cos γ
Relation between direction cosines is l2 + m2 + n2 = 1

Direction Ratios :
An ordered triple of numbers proportional to the direction cosines of a line are defined as ‘Direction ratios’ of that fine.

→ If \(\frac{1}{a}=\frac{m}{b}=\frac{n}{c}\) then a, b, c are called direction ratios of the line.

→ If a, b, c are the direction ratios of a line, then its direction cosines are \(\left(\frac{a}{\sqrt{a^{2}+b^{2}+c^{2}}}, \frac{b}{\sqrt{a^{2}+b^{2}+c^{2}}}, \frac{c}{\sqrt{a^{2}+b^{2}+c^{2}}}\right)\)

→ Direction ratios of the line joining P(x1, y1, z1) and Q (x2, y2, z2) are (x2 – x1, y2 – y1, z2 – z1) or (x1 – x2, y1 – y2, z1 – z2)

→ D.cs of the above line \(\frac{x_{2}-x_{1}}{P Q}, \frac{y_{2}-y_{1}}{P Q}, \frac{z_{2}-z_{1}}{P Q}\) or \(\frac{x_{1}-x_{2}}{P Q}, \frac{y_{1}-y_{2}}{P Q}, \frac{z_{1}-z_{2}}{P Q}\)

→ Angle between the lines whose D.cs are (l1 m1, n1) and (l2, m2, n2) is given by cos θ = l1l2 + m1m2 + n1n2

→ If these lines are perpendicular, then l1l2 + m1m2 + n1n2 = 0

Inter 1st Year Maths 1B Direction Cosines and Direction Ratios Formulas

Angle Between Two Lines:
The angle between two skew lines is the angle between two lines drawn parallel to them through any point in space.
Inter 1st Year Maths 1B Direction Cosines and Direction Ratios Formulas 1

Direction Cosines:
If α, β, γ are the angles made by a directed line segment with the positive directions of the coordinate axes respectively, then cos α, cos β, cos γ are called the direction cosines of the given line and they are denoted by l, m, n respectively Thus l = cos α, m = cos β, n = cos γ
Inter 1st Year Maths 1B Direction Cosines and Direction Ratios Formulas 2

The direction cosines of \(\overline{o p}\) are l = cos α, m = cos β, n = cos γ.
If l, m, n are the d.c’s of a line L is one direction then the d.c’s of the same line in the opposite direction are -l, -m, -n.

Note :

  • The angles α, β, γ are known as the direction angles and satisfy the condition 0 < α, β, γ < π.
  • The sum of the angles α, β, γ is not equal to 2β because they do not lie in the same plane.
  • Direction cosines of coordinate axes.

The direction cosines of the x-axis are cos0, cos\(\frac{\pi}{2}\), cos\(\frac{\pi}{2}\) i.e., 1, 0, 0
Similarly the direction cosines of the y-axis are (0, 1, 0) and z-axis are (0, 0, 1)

Theorem:
If P(x, y, z) is any point in space such that OP = r and if l, m, n are direction cosines of \(\overline{O P}\) then x = lr,y = mr, z = nr.
Note: If P(x, y, z) is any point in space such that OP = r then the direction cosines of \(\overline{O P}\) are
Note: If P is any point in space such that OP =r and direction cosines of OP are l,m,n then the point P =(lr,mr,nr)
Note: If P(x,y,z) is any point in space then the direction cosines of OP are \(\frac{x}{\sqrt{x^{2+} y^{2}+z^{2}}}, \frac{y}{\sqrt{x^{2+} y^{2}+z^{2}}}, \frac{z}{\sqrt{x^{2+} y^{2}+z^{2}}}\)

Theorem:
If l, m, n are the direction cosines of a line L then l2 + m2 + n2 = 1.
Proof:
Inter 1st Year Maths 1B Direction Cosines and Direction Ratios Formulas 3
l = cos α = \(\frac{x}{r}\), m = cos β = \(\frac{y}{r}\), n = cos γ = \(\frac{z}{r}\) ⇒ cos2α + cos2β + cos2γ = \(\frac{x^{2}}{r^{2}}+\frac{y^{2}}{r^{2}}+\frac{z^{2}}{r^{2}}\)
= \(\frac{x^{2}+y^{2}+z^{2}}{r^{2}}=\frac{r^{2}}{r^{2}}\)
∴ l2 + m2 + n2 = 1

Theorem:
The direction cosines of the line joining the points P(x1, y1, z1),Q(x2, y2, z2) are
\(\left(\frac{x_{2}-x_{1}}{r}, \frac{y_{2}-y_{1}}{r}, \frac{z_{2}-z_{1}}{r}\right)\) where r = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}\)

Direction Ratios:
A set of three numbers a,b,c which are proportional to the direction cosines l,m,n respectively are called DIRECTION RATIOS (d.r’s) of a line.
Note : If (a, b, c) are the direction ratios of a line then for any non-zero real number λ , (λa, λb, λc) are also the direction ratios of the same line.
Direction cosines of a line in terms of its direction ratios
If (a, b, c) are direction ratios of a line then the direction cosines of the line are ±\(\left(\frac{a}{\sqrt{a^{2}+b^{2}+c^{2}}}, \frac{b}{\sqrt{a^{2}+b^{2}+c^{2}}}, \frac{c}{\sqrt{a^{2}+b^{2}+c^{2}}}\right)\)

Theorem:
The direction ratios of the line joining the points are (x2 – x1, y2 – y1, z2 – z1)

Inter 1st Year Maths 1B Direction Cosines and Direction Ratios Formulas

Angle Between Two Lines:
If (l1, m1, n1) and (l2, m2, n2) are the direction cosines of two lines θ and is the acute angle between them, then cos θ = |l1l2 + m1m2 + n1n2|

Note.
If θ is the angle between two lines having d.c’s (l1, m1, n1) and (l2, m2, n2) then
sin θ = \(\sqrt{\sum\left(l_{1} m_{2}-l_{2} m_{1}\right)^{2}}\) and tan θ = \(\frac{\sqrt{\sum\left(l_{1} m_{2}-l_{2} m_{1}\right)^{2}}}{\left|l_{1} l_{2}+m_{1} m_{2}+n_{1} n_{2}\right|}\) where θ ≠ \(\frac{\pi}{2}\)

Note :
The condition for the lines to be perpendicular is l1l2 + m1m2 + n1n2 = 0
The condition for the lines to be parallel is \(\frac{l_{1}}{l_{2}}=\frac{m_{1}}{m_{2}}=\frac{n_{1}}{n_{2}}\)

Theorem:
If (a1, b1, c1) and (a2, b2, c2) are direction ratios of two lines and θ is the angle
between them then cos θ = \(\frac{a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}} \sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\)

Note:

  • If the two lines are perpendicular then a1a2 + b1b2 + c1c2 = 0
  • If the two lines are parallel then \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)
  • If one of the angle between the two lines is 0 then other angle is 180° – θ