Example.1
In a quadrilateral PQRS
,
,
, M is the midpoint
of
and X is a
point on SM such that SX =
SM. Then prove that P, X and
R are collinear.
Solution
![](http://www.quizsolver.com/radix/dth/notif/VECTOR_2_SUB_files/image006.gif)
&⇒ ![](http://www.quizsolver.com/radix/dth/notif/VECTOR_2_SUB_files/image007.gif)
![](http://www.quizsolver.com/radix/dth/notif/VECTOR_2_SUB_files/image008.gif)
![](http://www.quizsolver.com/radix/dth/notif/VECTOR_2_SUB_files/image010.gif)
![](http://www.quizsolver.com/radix/dth/notif/VECTOR_2_SUB_files/image011.gif)
![](http://www.quizsolver.com/radix/dth/notif/VECTOR_2_SUB_files/image012.gif)
also ![](http://www.quizsolver.com/radix/dth/notif/VECTOR_2_SUB_files/image013.gif)
&⇒
,
hence P, X and R are collinear.
Example.2
Unit vectors
are perpendicular to each other
and the unit vector
is inclined at an angle
q to both
. If
are
real, prove that ![](http://www.quizsolver.com/radix/dth/notif/VECTOR_2_SUB_files/image018.gif)
Solution
![](http://www.quizsolver.com/radix/dth/notif/VECTOR_2_SUB_files/image020.gif)
Example.3
Given that
are
the position vectors of points P and Q respectively. Find the equation for the
plane passing through Q and perpendicular to the line PQ. What is the distance
from the point (-1, 1, 1) to the plane?
Solution
![](http://www.quizsolver.com/radix/dth/notif/VECTOR_2_SUB_files/image022.gif)
=
= - ![](http://www.quizsolver.com/radix/dth/notif/VECTOR_2_SUB_files/image024.gif)
Equation of plane
passing through Q and perpendicular to PQ is
![](http://www.quizsolver.com/radix/dth/notif/VECTOR_2_SUB_files/image025.gif)
&⇒ ![](http://www.quizsolver.com/radix/dth/notif/VECTOR_2_SUB_files/image026.gif)
&⇒ ![](http://www.quizsolver.com/radix/dth/notif/VECTOR_2_SUB_files/image027.gif)
&⇒
…(1)
Let ![](http://www.quizsolver.com/radix/dth/notif/VECTOR_2_SUB_files/image029.gif)
Hence distance from
to the plane (1) is
=
=
=
5 units
Example.4
Find the vector
which
makes equal angles with the vectors
and is
perpendicular to the vector
= (1,-1,2)
with
and the angle between
and the unit vector
is obtuse.
Solution
….(1)
is perpendicular to
![](http://www.quizsolver.com/radix/dth/notif/VECTOR_2_SUB_files/image041.gif)
&⇒![](http://www.quizsolver.com/radix/dth/notif/VECTOR_2_SUB_files/image042.gif)
&⇒ x–y+2z =0 …(2)
Also it is given
that ,
(
makes
equal angles with
)
Since![](http://www.quizsolver.com/radix/dth/notif/VECTOR_2_SUB_files/image046.gif)
We get
&⇒ xy – 2yz +3xz = 2xz +3xy– yz
&⇒ 2xy+yz– zx = 0 ….(3).
As the angle between
is given to be obtuse.
….(4)
from (2) & (3)
we get, 2y (y – 2z) + yz – z(y – 2z) = 0
&⇒ (y – z)2 = 0,y= z and x
= –z
From (1) get z2
+ z2 + z2 = 12
&⇒
z2 = 4
\ z = –2 (z = y < 0)
Hence
Example.5
Show that midpoints of the
three diagonals of a complete quadrilateral are collinear . .
Solution
A complete quadrilateral
is a figure made by four straight lines no three of which are
concurrent. .
Let ABCD be a quadrilateral and let P, Q and R be the
respective midpoints of the diagonals BD, CA and EF. .
![](http://www.quizsolver.com/radix/dth/notif/VECTOR_2_SUB_files/image051.gif)
Let
be the position vectors of A,
B, C, D, E, F, P, Q & R respectively.
then
,
and ![](http://www.quizsolver.com/radix/dth/notif/VECTOR_2_SUB_files/image055.gif)
since F lies on the two lines BC and AD
and E lies on the two lines BA and CD respectively, we have
. . . (1)
.
. . (2)
. . .
(3)
. . . (4)
where s, t , u and v
are scalars
adding (1) , (2), (3)
and (4) , we get
(2 – s – u)(
) + (1 – t + u)(
) - 2(
) = 0
setting 1 – t + u = 1 + s
– u and 2 – s – v =t + v
so that (2 – s – u) + (1
– t + u) –2 = 0
This shows that the points P, Q and
R are collinear since the sum (2 – s - u) + ( 1 – t + u) – 2= 0 . .