CHECK THE POINTS ARE VERTICES OF ISOSCELES TRIANGLE IN VECTOR

Example 1 :

Show that the points A (1, 1, 1), B(1, 2, 3) and C(2, - 1, 1) are vertices of an isosceles triangle

Solution :

In an isosceles triangle length of two sides will be equal. 

OA vector  =  (i + j + k) vector

OB vector  =  (i + 2j + 3k) vector

OC vector  =  (2i - j + k) vector

AB vector  =  OB - OA

  =  (i + 2j + 3k) - (i + j + k)

  =  (j + 2k) vector

|AB|  =  √12 + 2=  √5

BC vector  =  OC - OB

  =  (2i - j + k)  -  (i + 2j + 3k) 

  =  (i - 3j - 2k) vector

|BC|  =  √12 + (-3)2(-2)2    =  √14

CA vector  =  OC - OB

  =  (2i - j + k) -  (i + j + k) 

  =  (i - 2j) vector

|CA|  =  √12 + (-2)2   =  √5

Sides AB and CA are equal. Hence the given points are the vertices of isosceles triangle.

Example 2 :

Find the value or values of m for which m (i + j + k) is a unit vector.

Solution :

a vector  =  mi vector + mj vector + mk vector

If the given is unit vector, then |a vector|  =  1

  √m2 + m+ m=  1

  √3m2  =  1

m2  =  1/3

m  =  ±1/√3

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