p= i-2j+2k and Q=3i-j+k. find the vector which when added to ( P-Q) will give a unit vector along the z-axis
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Answer:
Let required vector be a
i
∧
+b
j
∧
+c
k
∧
∵ It makes equal angle with x and y-axis.
⟹
a
2
+b
2
+c
2
a
=
a
2
+b
2
+c
2
b
⟹a=b
∵It is a unit vector.
⟹a
2
+b
2
+c
2
=1−(i)
and it is orthogonal to −
i
∧
+2
j
∧
+2
k
∧
⟹−a+2b+2c=0
∵a=b
⟹−a+2a+2c=0
⟹2c=−a
⟹c=
2
−a
Puttingb=a,c=
2
−a
in(i):
⟹a
2
+a
2
+
4
a
2
=1
⟹4a
2
+4a
2
+a
2
=4
⟹9a
2
=4
⟹a=±
3
2
∴a
i
∧
+b
j
∧
+c
k
∧
=a
i
∧
+a
j
∧
+
2
−a
k
∧
=±
3
1
(2
i
∧
+2
j
∧
−
k
∧
)
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