Scalar (or Dot) Product of two vectors:
The scalar product of
and
,
written as
, is defined to be the
cosq
where q is the angle
between the vectors
and
i.e.
= ab cos q
![](http://www.quizsolver.com/radix/dth/notif/VECTOR_4_files/image007.gif)
Geometrical Interpretation
is
the product of length of one vector and length of the projection
of the other vector in the direction of former.
.
Projection of b in direction of
=
.Projection of a in direction of ![](http://www.quizsolver.com/radix/dth/notif/VECTOR_4_files/image006.gif)
So projection of
on
=
.
Properties
![](http://www.quizsolver.com/radix/dth/notif/VECTOR_4_files/image012.gif)
&⇒ ![](http://www.quizsolver.com/radix/dth/notif/VECTOR_4_files/image013.gif)
![](http://www.quizsolver.com/radix/dth/notif/VECTOR_4_files/image014.gif)
![](http://www.quizsolver.com/radix/dth/notif/VECTOR_4_files/image015.gif)
(acosq)b
= (projection of
) b = (projection of
) a
(Distributive law)![](http://www.quizsolver.com/radix/dth/notif/VECTOR_4_files/image020.gif)
Û
are perpendicular to each other ( if )![](http://www.quizsolver.com/radix/dth/notif/VECTOR_4_files/image024.gif)
= ![](http://www.quizsolver.com/radix/dth/notif/VECTOR_4_files/image026.gif)
- If
then![](http://www.quizsolver.com/radix/dth/notif/VECTOR_4_files/image029.gif)
- If
are non-zero, the
angle between them is given by ![](http://www.quizsolver.com/radix/dth/notif/VECTOR_4_files/image031.gif)
- =
![](http://www.quizsolver.com/radix/dth/notif/VECTOR_4_files/image032.gif)
Example.1
If
are
mutually perpendicular vectors of equal magnitude show that
are equally inclined to
.
Solution
We have
=
×
![](http://www.quizsolver.com/radix/dth/notif/VECTOR_4_files/image037.gif)
=
= 3|
|2
&⇒
= ![](http://www.quizsolver.com/radix/dth/notif/VECTOR_4_files/image040.gif)
Now
×
= |
|2 =
cosq1
where q1 is the angle between
and
.
Hence |
|2
=
cosq1
&⇒
cosq1 = ![](http://www.quizsolver.com/radix/dth/notif/VECTOR_4_files/image044.gif)
Similarly, if q2, q3 are the angles between
and
, then cosq2 =
, cosq3=
.
Hence q1 = q2 = q3.
Example.2
Find the unit vector which makes angle of
45° with the vector 2i + 2j – k and an angle of 60° with the vector
.
Solution
Let the unit vector be
so
that ![](http://www.quizsolver.com/radix/dth/notif/VECTOR_4_files/image048.gif)
&⇒
and ![](http://www.quizsolver.com/radix/dth/notif/VECTOR_4_files/image050.gif)
&⇒c2 – c3 = ![](http://www.quizsolver.com/radix/dth/notif/VECTOR_4_files/image051.gif)
&⇒
and ![](http://www.quizsolver.com/radix/dth/notif/VECTOR_4_files/image053.gif)
Also ![](http://www.quizsolver.com/radix/dth/notif/VECTOR_4_files/image054.gif)
&⇒
=
1
&⇒ c3 =![](http://www.quizsolver.com/radix/dth/notif/VECTOR_4_files/image056.gif)
Hence the required vectors are
.
Example.3
Find the component of
in the direction of the vector ![](http://www.quizsolver.com/radix/dth/notif/VECTOR_4_files/image059.gif)
Solution
A unit vector along
is
![](http://www.quizsolver.com/radix/dth/notif/VECTOR_4_files/image061.gif)
Hence the component of
along the given vector
= Projection of
on
=
=
![](http://www.quizsolver.com/radix/dth/notif/VECTOR_4_files/image064.gif)
Example.4
Show that any two opposite edges of a regular
tetrahedron are perpendicular.
Note: (A tetrahedron whose edges are all
equal in length is called a regular tetrahedron)..
Solution
Take
one of the vertices of the tetrahedron as the initial point. Let it be O. Then
the position vectors of the points A, B, C of the tetrahedron OABC are
, where |
| =
|
| = |
|.
Since all the edges are of equal length, all the four faces of the tetrahedron
are equilateral triangles
&⇒
![](http://www.quizsolver.com/radix/dth/notif/VECTOR_4_files/image067.gif)
&⇒
= 0. Hence the opposite edges are
perpendicular to each other.
Example.5
are three coplanar
vectors. If is not parallel to , show that.
Solution
Since
are
coplanar, we may write ![](http://www.quizsolver.com/radix/dth/notif/VECTOR_4_files/image074.gif)
&⇒
= l1
+ l2
…(1)
and
= l1
+ l2
…(2)
Solving (1) and (2), by Cramer’s rule, we
find that
and ![](http://www.quizsolver.com/radix/dth/notif/VECTOR_4_files/image082.gif)
Substituting for l1 and l2, we get the
required expression. .