Question

The measure acute angle between the lines whose direction ratios are 3,2, 6 and -2,1,2 is

A)cos-1(1/7). B)cos-1(8/15)
C)cos-1(1/3). D)cos-1(8/21)

Answer 1

Direction ratios of two lines are
For first line : a₁ = 3, b₁ = 2 , c₁ = 6
For 2nd line : a₂ = -2, b₂ = 1 , c₂ = 2

So, angle between two lines when direction ratios are (a₁,b₁,c₁) and (a₂,b₂,c₂) is given by cosθ = $\bold{\frac{|a_1.a_2+b_1.b_2+c_1.c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}}$
so, cosθ = (3 × -2 + 2 × 1 + 6 × 2 )/√(3² + 2² + 6²)√{(-2)²+1²+(2)²}
= (-6 + 2 + 12)/√(49).√(9) = 8/7 × 3 = 8/21
cosθ = 8/21 ⇒θ = cos⁻¹(8/21)

Hence, option ( D) is correct.