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Joint Entrance Examination
Mathematics
3D Geometry
Question 188
Question 188 of 521
Easy
MCQ
Let
P
(
3
,
2
,
3
)
,
Q
(
4
,
6
,
2
)
,
R
(
7
,
3
,
2
)
P(3,2,3),\ Q(4,6,2),\ R(7,3,2)
P
(
3
,
2
,
3
)
,
Q
(
4
,
6
,
2
)
,
R
(
7
,
3
,
2
)
be the vertices of
△
P
Q
R
\triangle PQR
△
P
QR
. Then the angle
∠
Q
P
R
\angle QPR
∠
QP
R
is:
A
π
6
\dfrac{\pi}{6}
6
π
B
cos
−
1
(
7
18
)
\cos^{-1}\left(\dfrac{7}{18}\right)
cos
−
1
(
18
7
)
C
cos
−
1
(
1
18
)
\cos^{-1}\left(\dfrac{1}{18}\right)
cos
−
1
(
18
1
)
D
π
3
\dfrac{\pi}{3}
3
π
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Official solution
\[\vec{PQ}=(1,4,-1),\qquad \vec{PR}=(4,1,-1).\] \[\cos(\angle QPR)=\frac{\vec{PQ}\cdot\vec{PR}}{|\vec{PQ}||\vec{PR}|}=\frac{4+4+1}{18}=\frac12.\] \[\angle QPR=\frac{\pi}{3}.\]
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