Question

Show that the vectors form the vertices of a right angled triangle.

Answer 1

The given vectors are 2 i ^ − j ^ + k ^ , i ^ −3 j ^ −5 k ^ and 3 i ^ −4 j ^ −4 k ^ .

Let vectors 2 i ^ − j ^ + k ^ , i ^ −3 j ^ −5 k ^ and 3 i ^ −4 j ^ −4 k ^ represent the position vector of points A, B and C.

Condition for right angled triangle is: | AC → | 2 + | BC → | 2 = | AB | → 2 (1)

OA → =2 i ^ − j ^ + k ^ , OB → =2 i ^ − j ^ + k ^ and OC → =3 i ^ −4 j ^ −4 k ^

AB → =( 1−2 ) i ^ +( −3+1 ) j ^ +(−5−1) k ^ =− i ^ +−2 j ^ −6 k ^

| AB | → = ( −1 ) 2 + ( −2 ) 2 + ( −6 ) 2 = 1+4+36 = 41 (2)

BC → =( 3−1 ) i ^ +( −4+3 ) j ^ +(−4+5) k ^ =2 i ^ +− j ^ + k ^

| BC | → = 2 2 + ( −1 ) 2 + (1) 2 = 4+1+1 = 6 (3)

AC → =( 2−3 ) i ^ +( −1+4 ) j ^ +(1+4) k ^ =− i ^ +3 j ^ +5 k ^

| AC | → = ( −1 ) 2 + 3 2 + 5 2 = 1+9+25 = 35 (4)

From (2), (3) and (4), we get,

Since, | AC → | 2 + | BC → | 2 = | AB | → 2 Using (1).

Thus, the vectors 2 i ^ − j ^ + k ^ , i ^ −3 j ^ −5 k ^ and 3 i ^ −4 j ^ −4 k ^ are the vertices of right angled triangle.