Question

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

Answer 1

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 ^ ) =( 2−3 ) 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 ^ ) =( 1−2 ) i ^ +( −3+1 ) j ^ +( −5−1 ) 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 ^ ) =( 3−1 ) 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.