
Vectors in a Plane and Space 

Vector
product or cross product 
The mixed product
or the scalar triple product 
The vector product
and the mixed product use, examples 





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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