
Vectors in a Plane and Space 

The mixed product
or the scalar triple product 
The mixed product
or scalar triple product definition 
The
mixed product properties 
The condition for
three vectors to be coplanar 
The mixed product
or scalar triple product expressed in terms of components 
The vector product
and the mixed product use, examples 






The mixed product 
The
mixed product or scalar triple product definition

The
mixed product (or the scalar triple product) is the scalar product of the first vector with the vector product

of the other two
vectors denoted as a
· ( b ´
c ).

Geometrically, the
mixed product is the volume of a parallelepiped defined by vectors,
a ,
b
and c
as shows the right figure.

The vector b
´
c is perpendicular to the base of the
parallelepiped and its magnitude equals the area of
the

base, B
=  b ´
c
.

The altitude of the parallelepiped is projection of the
vector a in the direction of
the vector
b ´
c,
so

h =
 a  · cos j.




Therefore, the scalar product of the vector
a
and vector
b ´
c
is equal to the volume of the parallelepiped.


The
sign of the scalar triple product can be either positive or negative, as
a
· ( b ´
c ) =

a
· ( c ´
b ).


The
mixed product properties 


The condition for
three vectors to be coplanar 
The
mixed product is zero if any two of vectors, a ,
b
and c
are parallel, or if a ,
b
and c
are coplanar. That is, when the given three vectors are coplanar the altitude of the parallelepiped is zero and thus the scalar triple product is zero, 


The
mixed product or scalar triple product expressed in terms of components 
The scalar triple product expressed in terms of the components of vectors,
a =
a_{x}i + a_{y }j + a_{z}k, 
b
= b_{x}i + b_{y }j +
b_{z}k
and c
= c_{x}i + c_{y }j +
c_{z}k, 

The above formula can be derived from the determinant expanded by minors through the elements of the
first row, 

Therefore, vectors,
a,
b
and c
will be coplanar if the determinant is zero. 

The vector product
and the mixed product use examples 
Example:
Given are vectors, a
= i 
2_{ }k and
b =

i + 3_{ }j + k, determine the vector
c
= a
´
b
and the area of a parallelogram formed by vectors, a
and b.

Solution: 



The area of the
parallelogram, 

or 






Example: Vertices of a triangle are, A(1,
0, 1), B(3,
2,
0) and C(4,
1, 2),
find the length of the altitude h_{b}.

Solution:
Using the area of
a triangle,


as
the area of the triangle


where 






then,
the area of the triangle, 



and the
length of the side b, 




thus,
the length of the altitude, 






Example: Find the angles that
the unit vector, which is orthogonal to the plane formed by vectors,

a
= i +
2_{ }j + k
and b
= 3i 
2_{}_{
}j + 4k,
makes with the coordinate axes. 
Solution:
The unit vector which is orthogonal to the plane,
formed by the vectors, a
and b,
is the direction vector of a vector c,
such that





Angles, which the unit vector
c°
forms with coordinate axes, we find by using the formula, 


Example: Given are vectors,
a
= 2i 
3_{ }j,
b
= i 
2_{
}j + 3k
and c
= i +
2_{
}j + k,
find the projection
(the scalar component) of the vector a
onto vector d
= b ´
c.

Solution:
Let first find
the vector d, 



Example: Given are vectors,
a
= 3i +
4_{ }j 
l_{
}k,
b
=
2i  _{
}j + k
and c
= i

4_{ }j 3l
k,
determine the
parameter l
such that vectors to be coplanar. 
Solution:
Vectors lie on the same plane if their scalar triple product is zero, i.e.,
V =
0, therefore vectors’
coordinates must satisfy the condition, 


Example: Examine if vectors,
a
= 4i + 2_{ }j +
k,
b
= 3i + 3_{ }j 
2k
and c
= 
5i
 _{
}j 
4k,
are coplanar and if so,
prove their linear dependence. 
Solution:
Check for coplanarity, 

Since vectors are coplanar each of them can be represented as linear combination of other two. 


Example: Prove that points,
A(2,
1, 0), B(3,
2,
1), C(1,
1,
2) and D(3,
2, 5),
all lie on the same plane. 
Solution:
If all given points belong to the same plane then vectors,
AB,
AC
and AD
are coplanar, therefore the scalar triple product (AB ´
AC)
· AD
= 0. 


thus, all
given points lie on the same plane. 








Vectors
in 2D and 3D Contents 



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