The angle between the lines \( \displaystyle\frac{3-x}{-7}=\frac{y+5}{-5}=\frac{z}{1} \) and \( \displaystyle\frac{x}{1}=\frac{y}{2}=\frac{z}{3} \) is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → π/2
The first line is not in standard form, because x has coefficient −1 in the numerator. Rewriting it:
\( \displaystyle \frac{3-x}{-7}=\frac{x-3}{7} \)
\( \displaystyle \frac{x-3}{7}=\frac{y+5}{-5}=\frac{z}{1} \)
So the direction ratios are 7, −5, 1 for the first line and 1, 2, 3 for the second.
\( \displaystyle \vec{b}_1\cdot\vec{b}_2=7(1)+(-5)(2)+1(3) \)
\( \displaystyle =7-10+3=0 \)
The dot product is zero, so the lines are perpendicular and the angle is π/2.
Tip: Direction ratios are read only after every variable appears as x − x1, y − y1, z − z1; taking −7 here would give a non-zero dot product and a wrong angle.