Find the angle between the lines whose direction ratios are proportional to a,b,c and b–c, c–a, a–b.
Let us assume the direction ratios of vectors be (r1,r2,r3) and (r4,r5,r6).
Then,
⇒ (r1,r2,r3) = (a,b,c)
⇒ (r4,r5,r6) = (b–c, c–a, a–b)
We know that the angle between the lines with direction ratios proportional to (a1,b1,c1) and (a2,b2,c2) is given by:
⇒
Using the above formula we calculate the angle between the lines.
Let be the angle between the two lines given in the problem.
⇒
⇒
⇒
⇒
⇒
∴ The angle between two given vectors is .