# If OAaandOBbOA¯=a¯andOB¯=b¯, then show that the vector along the angle bisector of ∠AOB is given by daabbd¯=λ(a¯|a¯|+b¯|b¯|). - Mathematics and Statistics

Sum

If bar"OA" = bar"a" and bar"OB" = bar"b", then show that the vector along the angle bisector of ∠AOB is given by bar"d" = lambda(bar"a"/|bar"a"| + bar"b"/|bar"b"|).

#### Solution

Choose any point P on the angle bisector of ∠AOB. Draw PM parallel to OB.

∴ ∠OPM = ∠POM = ∠POB

Hence, OM = MP

∴ OM and MP is the same scalar multiple of unit vectors hat"a" and hat"b" along these directions,

where hat"a" = bar"a"/|bar"a"| and hat"b" = bar"b"/|bar"b"|

∴ bar"OM" = lambdahat"a" and bar"MP" = lambdahat"b"

∴ bar"OP" = bar"OM" + bar"MP"

= lambdahat"a" + lambdahat"b"

= lambda(hat"a" + hat"b")

Hence, the vector along angle bisector of ∠AOB is given by

bar"d" = bar"OP" = lambda(bar"a"/|bar"a"| + bar"b"/|bar"b"|)

Concept: Vectors and Their Types
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