STRAIGHT LINE IN SPACE

λ γ μ ₁₂₃₄ ₙ ₐ ₓ ˣⁱⁿ⁺ ⁻ ₊ ₋ 

1) Vector equation of a line:

The vector equation of a straight line passing through a fixed point with position vector a and parallel to a given vector b is 

r= a+ λb, where λ is scalar.

Note :

a) In the above equation r is the position vector of any point (x,y,z) on the line. Therefore, r= xi + yj+ zk.

b) The position vector of any point on the line is taken a+ λb.

2) The vector equation of a line passing through two points with position vector a and b is:

r= a + λ(b - a).

3) The Cartesian equation of a straight line passing through a fixed point (x₁ , y₁ , z₁) and having direction ratios proportional to a, b, c is:

(x - x₁)/a = (y - y₁)/b = (z - z₁)/c.

Note:

a) The above form of a line is known as the symmetrical form of a line.

b) The parametric equations of the line (x - x₁)/a = (y - y₁)/b = (z - z₁)/c. are x= x₁ + aλ, y= y₁ + b λ ,.z=  z₁ + cλ, where λ is the parameter.

c) The coordinates of any point on the line (x - x₁)/a = (y - y₁)/b = (z - z₁)/c are 

(x₁ + aλ,  y₁ + b λ , z₁ + cλ) where λ belongs R.

d) Since the direction cosines of a line are are its direction ratios. Therefore, equation of a line passing through (x₁ , y₁ , z₁) and having direction cosines l,.m n are

(x - x₁)/l = (y - y₁)/m = (z - z₁)/n

e) Since x, y and z-axes pass through the origin and have direction cosines 1,0,0; 0,1,0; and 0,0,1 respectively. Therefore, their equation are.