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Exams
Joint Entrance Examination
Mathematics
Vector Algebra
Question 23
Question 23 of 379
Medium
MCQ
Let
a
⃗
=
i
^
+
2
j
^
+
k
^
\vec{a}=\hat{i}+2\hat{j}+\hat{k}
a
=
i
^
+
2
j
^
+
k
^
and
b
⃗
=
2
i
^
+
j
^
−
k
^
\vec{b}=2\hat{i}+\hat{j}-\hat{k}
b
=
2
i
^
+
j
^
−
k
^
. Let
c
^
\hat{c}
c
^
be a unit vector in the plane of the vectors
a
⃗
\vec{a}
a
and
b
⃗
\vec{b}
b
and be perpendicular to
a
⃗
\vec{a}
a
. Then such a vector
c
^
\hat{c}
c
^
is:
A
\frac{1}{\sqrt{\sqrt{2}}}(-\overset{^}{i}+\overset{^}{k})
B
\frac{1}{\sqrt{\sqrt{5}}}(\overset{^}{j}-2\overset{^}{k})
C
\frac{1}{\sqrt{\sqrt{3}}}(\overset{^}{i}-\overset{^}{j}+\overset{^}{k})
D
\frac{1}{\sqrt{\sqrt{3}}}(-\overset{^}{i}+\overset{^}{j}-\overset{^}{k})
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Solutions · 1
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Official solution
Let vector
p
⃗
\vec{p}
p
in plane of
a
⃗
&
b
⃗
=
K
(
a
⃗
+
λ
b
⃗
)
\vec{a}\&\vec{b}=K(\vec{a}+\lambda \vec{b})
a
&
b
=
K
(
a
+
λ
b
)
p
→
⊥
a
→
=
p
→
⋅
a
→
=
0
⇒
K
(
a
→
+
λ
b
→
)
⋅
a
→
=
0
⇒
a
→
⋅
a
→
+
λ
b
→
⋅
a
→
=
0
⇒
6
+
λ
(
3
)
=
0
⇒
λ
=
−
2
⇒
p
→
=
(
−
3
i
^
+
3
k
^
)
\begin{gathered} \overset{\to }{p}\perp \overset{\to }{a}=\overset{\to }{p}\cdot \overset{\to }{a}=0 \\ \Rightarrow K(\overset{\to }{a}+\lambda \overset{\to }{b})\cdot \overset{\to }{a}=0 \\ \Rightarrow \overset{\to }{a}\cdot \overset{\to }{a}+\lambda \overset{\to }{b}\cdot \overset{\to }{a}=0 \\ \Rightarrow 6+\lambda (3)=0 \\ \Rightarrow \lambda =-2 \\ \Rightarrow \overset{\to }{p}=(-3\hat{i}+3\hat{k}) \end{gathered}
p
→
⊥
a
→
=
p
→
⋅
a
→
=
0
⇒
K
(
a
→
+
λ
b
→
)
⋅
a
→
=
0
⇒
a
→
⋅
a
→
+
λ
b
→
⋅
a
→
=
0
⇒
6
+
λ
(
3
)
=
0
⇒
λ
=
−
2
⇒
p
→
=
(
−
3
i
^
+
3
k
^
)
Unit vector
→
±
(
−
i
^
+
k
^
)
2
\to \pm \frac{(-\hat{i}+\hat{k})}{\sqrt{2}}
→
±
2
(
−
i
^
+
k
^
)
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