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Mathematics Question on angle between two lines

The angle between the lines 2x=3y=−z2x = 3y = - z and 6x=−y=−4z6x = - y = - 4z is

A

0∘0^{\circ}

B

45∘45^{\circ}

C

90∘90^{\circ}

D

30∘30^{\circ}

Answer

90∘90^{\circ}

Explanation

Solution

Given L1:2x=3y=−zL_1 : 2x = 3y = - z and L2:6x=−y=−4xL_2 : 6x = - y = -4x
⇒ L1:x12=y13=z−1\Rightarrow \, L_1 : \frac{x}{\frac{1}{2}} = \frac{y}{\frac{1}{3}} = \frac{z}{-1} and L2:x116=y−1=z−14L_2 : \frac{x}{\frac{1}{16}} = \frac{y}{-1} = \frac{z}{ - \frac{1}{4}}
Now, a1a2+b1b2+c1c2=(12×16)+(13×−1)+(−1×−14)=112−13+14=0a_1 a_2 + b_1 b_2 + c_1 c_2 = \left(\frac{1}{2} \times\frac{1}{6}\right) + \left(\frac{1}{3} \times-1\right) + \left(-1 \times- \frac{1}{4}\right) = \frac{1}{12} - \frac{1}{3} + \frac{1}{4}= 0
∴\therefore Angle between L1L_1 and L2L_2 is 90∘90^{\circ}