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Joint Entrance Examination
Mathematics
Vector Algebra
Question 55
Question 55 of 379
Medium
MCQ
Let a unit vector which makes an angle of
60
∘
{60}^{\circ }
60
∘
with
2
i
^
+
2
j
^
−
k
^
2\hat{i}+2\hat{j}-\hat{k}
2
i
^
+
2
j
^
−
k
^
and an angle of
45
∘
{45}^{\circ }
45
∘
with
i
^
−
k
^
\hat{i}-\hat{k}
i
^
−
k
^
be
C
⃗
\vec{C}
C
. Then
C
⃗
+
(
−
1
2
i
^
+
1
3
2
j
^
−
2
3
k
^
)
\vec{C}+(-\frac{1}{2}\hat{i}+\frac{1}{3\sqrt{2}}\hat{j}-\frac{\sqrt{2}}{3}\hat{k})
C
+
(
−
2
1
i
^
+
3
2
1
j
^
−
3
2
k
^
)
is:
A
-\frac{\sqrt{\sqrt{2}}}{3}\overset{^}{i}+\frac{\sqrt{\sqrt{2}}}{3}\overset{^}{j}+(\frac{1}{2}+\frac{2\sqrt{\sqrt{2}}}{3})\overset{^}{k}
B
(\frac{1}{\sqrt{\sqrt{3}}}+\frac{1}{2})\overset{^}{i}+(\frac{1}{\sqrt{\sqrt{3}}}-\frac{1}{3\sqrt{\sqrt{2}}})\overset{^}{j}+(\frac{1}{\sqrt{\sqrt{3}}}+\frac{\sqrt{\sqrt{2}}}{3})\overset{^}{k}
C
\frac{\sqrt{\sqrt{2}}}{3}\overset{^}{i}-\frac{1}{2}\overset{^}{k}
D
\frac{\sqrt{\sqrt{2}}}{3}\overset{^}{i}+\frac{1}{3\sqrt{\sqrt{2}}}\overset{^}{j}-\frac{1}{2}\overset{^}{k}
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Official solution
C
⃗
=
a
i
^
+
b
j
^
+
c
k
^
(
a
i
^
+
b
j
^
+
c
k
^
)
⋅
(
2
i
^
+
2
j
^
−
k
^
)
=
1
×
3
×
1
2
2
a
+
2
b
−
c
=
3
2
(
a
i
^
+
b
j
^
+
c
k
^
)
⋅
(
i
^
−
k
^
)
=
1
×
2
×
1
2
a
−
c
=
1
a
2
+
b
2
+
c
2
=
1
\begin{gathered} \vec{C}=a\hat{i}+b\hat{j}+c\hat{k} \\ (a\hat{i}+b\hat{j}+c\hat{k})\cdot (2\hat{i}+2\hat{j}-\hat{k})=1\times 3\times \frac{1}{2} \\ 2a+2b-c=\frac{3}{2}\, \\ (a\hat{i}+b\hat{j}+c\hat{k})\cdot (\hat{i}-\hat{k})=1\times \sqrt{2}\times \frac{1}{\sqrt{2}} \\ a-c=1\, \\ {a}^{2}+{b}^{2}+{c}^{2}=1\, \end{gathered}
C
=
a
i
^
+
b
j
^
+
c
k
^
(
a
i
^
+
b
j
^
+
c
k
^
)
⋅
(
2
i
^
+
2
j
^
−
k
^
)
=
1
×
3
×
2
1
2
a
+
2
b
−
c
=
2
3
(
a
i
^
+
b
j
^
+
c
k
^
)
⋅
(
i
^
−
k
^
)
=
1
×
2
×
2
1
a
−
c
=
1
a
2
+
b
2
+
c
2
=
1
Solving (1), (2) and (3)
a
+
2
b
=
1
2
a
2
+
b
2
+
(
a
−
1
)
2
=
1
2
a
2
−
2
a
+
b
2
=
0
2
a
2
−
2
a
+
(
2
a
−
1
4
)
2
=
0
32
a
2
−
32
a
+
4
a
2
−
4
a
+
1
=
0
36
a
2
−
36
a
+
1
=
0
a
=
36
±
(
36
)
2
−
4
(
36
)
2
×
36
=
1
2
±
2
3
b
=
1
−
2
a
4
⇒
b
=
1
±
2
2
3
−
1
4
=
∓
1
3
2
C
=
−
1
2
±
2
3
\begin{gathered} a+2b=\frac{1}{2} \\ {a}^{2}+{b}^{2}+(a-1{)}^{2}=1 \\ 2{a}^{2}-2a+{b}^{2}=0 \\ 2{a}^{2}-2a+{(\frac{2a-1}{4})}^{2}=0 \\ 32{a}^{2}-32a+4{a}^{2}-4a+1=0 \\ 36{a}^{2}-36a+1=0 \\ a=\frac{36\pm \sqrt{(36{)}^{2}-4(36)}}{2\times 36} \\ =\frac{1}{2}\pm \frac{\sqrt{2}}{3} \\ b=\frac{1-2a}{4}\Rightarrow b=\frac{1\pm \frac{2\sqrt{2}}{3}-1}{4} \\ =\mp \frac{1}{3\sqrt{2}} \\ C=-\frac{1}{2}\pm \frac{\sqrt{2}}{3} \end{gathered}
a
+
2
b
=
2
1
a
2
+
b
2
+
(
a
−
1
)
2
=
1
2
a
2
−
2
a
+
b
2
=
0
2
a
2
−
2
a
+
(
4
2
a
−
1
)
2
=
0
32
a
2
−
32
a
+
4
a
2
−
4
a
+
1
=
0
36
a
2
−
36
a
+
1
=
0
a
=
2
×
36
36
±
(
36
)
2
−
4
(
36
)
=
2
1
±
3
2
b
=
4
1
−
2
a
⇒
b
=
4
1
±
3
2
2
−
1
=
∓
3
2
1
C
=
−
2
1
±
3
2
C
+
(
−
1
2
i
^
+
1
3
2
j
^
−
2
3
k
^
)
=
2
3
i
^
−
1
2
k
^
\begin{gathered} C+(\frac{-1}{2}\hat{i}+\frac{1}{3\sqrt{2}}\hat{j}-\frac{\sqrt{2}}{3}\hat{k}) \\ =\frac{\sqrt{2}}{3}\hat{i}-\frac{1}{2}\hat{k} \end{gathered}
C
+
(
2
−
1
i
^
+
3
2
1
j
^
−
3
2
k
^
)
=
3
2
i
^
−
2
1
k
^
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