Find the acute angle between two straight lines whose direction ratios

1.	Find the acute angle between two straight lines whose direction ratios are (1,1,0) and (2,1,2).
Answer» Let L₁ direction ratio is a = 1, b = 1, c = 0 and L₂ direction ratio a₂ = 2, b₂ = 1, c₂ = 2 Hence line L₁ and L₂ has acute angle, cos θ = \|(a₁a₂ + b₁b₂ + c₁c₂)/(√(a²₁ + b²₁ + c²₁) x √(a²₂ + b²₂ + c²₂))\| cos θ = \|(1 x 2 + 1 x 1 + 0 x 2)/(√(1² + 1² + 0²) x √(2² + 1² + 2²))\| cos θ = \|(2 + 1 + 0)/√(1 + 1) x √(4 + 1 + 4)\| cos θ = \|3/(√2 x √9)\| cos θ = \|3/3√2\| cos θ = 1/√2 cos θ = cos π/4 θ = π/4

Find the acute angle between two straight lines whose direction ratios are (1,1,0) and (2,1,2).

Answer»

Let L₁ direction ratio is a = 1, b = 1, c = 0 and L₂ direction ratio a₂ = 2, b₂ = 1, c₂ = 2

Hence line L₁ and L₂ has acute angle,

cos θ = |(a₁a₂ + b₁b₂ + c₁c₂)/(√(a²₁ + b²₁ + c²₁) x √(a²₂ + b²₂ + c²₂))|

cos θ = |(1 x 2 + 1 x 1 + 0 x 2)/(√(1² + 1² + 0²) x √(2² + 1² + 2²))|

cos θ = |(2 + 1 + 0)/√(1 + 1) x √(4 + 1 + 4)|

cos θ = |3/(√2 x √9)|

cos θ = |3/3√2|

cos θ = 1/√2

cos θ = cos π/4

θ = π/4