Question

prove that the vectors A=2i + j+ k ,B= 2i + j+ 2k and C= i + j+ k Can form the Sides of a right angled triangle​

Answer 1

Explanation:

Let vectors 2

i

^

−

j

^

+

k

^

,

i

^

−3

j

^

−5

k

^

and 3

i

^

−4

j

^

−4

k

^

be position vectors of points A,B and C respectively.

i.e.,

OA

=2

i

^

−

j

^

+

k

^

,

OB

=

i

^

−3

j

^

−5

k

^

and

OC

=3

i

^

−4

j

^

−4

k

^

Now, vectors

AB

,

BC

, and

AC

represent the sides of ΔABC.

∴

AB

=(1−2)

i

^

+(−3+1)

j

^

+(−5−1)

k

^

=−

i

^

−2

j

^

−6

k

^

BC

=(3−1)

i

^

+(−4+3)

j

^

+(−4+5)

k

^

=2

i

^

−

j

^

+

k

^

AC

=(2−3)

i

^

+(−1+4)

j

^

+(1+4)

k

^

=−

i

^

+3

j

^

+5

k

^

∣

AB

∣=

(−1)

2

+(−2)

2

+(−6)

2

=

1+4+36

=

41

∣

BC

∣=

2

2

+(−1)

2

+1

2

=

4+1+1

=

6

∣

AC

∣=

(−1)

2

+3

2

+5

2

=

1+9+25

=

35

∴∣

BC

∣

2

+∣

AC

∣

2

=6+35=41=∣

AB

∣

2

Hence, ΔABC is a right-angled triangle.