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Question

Show that the points A, B and C with position vectors, , respectively form the vertices of a right angled triangle.

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Solution

The given position vectors of triangle ABC are a =3 i ^ 4 j ^ 4 k ^ , b =2 i ^ 1 j ^ +1 k ^ and c =1 i ^ 3 j ^ 5 k ^ .

Vector representation of side AB is,

AB = b a =( 2 i ^ 1 j ^ +1 k ^ )( 3 i ^ 4 j ^ 4 k ^ ) =( 23 ) i ^ +( 1+4 ) j ^ +( 1+4 ) k ^ = i ^ +3 j ^ +5 k ^

Vector representation of side BC is,

BC = c b =( 1 i ^ 3 j ^ 5 k ^ )( 2 i ^ 1 j ^ +1 k ^ ) =( 12 ) i ^ +( 3+1 ) j ^ +( 51 ) k ^ = i ^ 2 j ^ 6 k ^

Vector representation of side CAis,

CA = a c =( 3 i ^ 4 j ^ 4 k ^ )( 1 i ^ 3 j ^ 5 k ^ ) =( 31 ) i ^ +( 4+3 ) j ^ +( 4+5 ) k ^ =2 i ^ 1 j ^ +1 k ^

Now,

| AB | 2 = ( 1 ) 2 + 3 2 + 5 2 =1+9+25 =35

And

| BC | 2 = ( 1 ) 2 + ( 2 ) 2 + ( 6 ) 2 =1+4+36 =41

And

| CA | 2 = ( 2 ) 2 + ( 1 ) 2 + ( 1 ) 2 =4+1+1 =6

By property of right angle triangle,

| AB | 2 + | CA | 2 =35+6 =41 = | BC | 2

Hence, ABC is a right angled triangle.


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