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
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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.
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