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Vectors in a Plane and Space |
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Vector
product or cross product |
The mixed product
or the scalar triple product |
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The vector product
and the mixed product use, examples |
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| 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.
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| Solution: |
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| The area of the
parallelogram, |
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| or |
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| 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 hb.
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Solution:
Using the area of
a triangle,
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| as
the area of the triangle
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where |
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| then,
the area of the triangle, |
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| and the
length of the side b, |
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| thus,
the length of the altitude, |
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| Example: Find the angles that
the unit vector, which is orthogonal to the plane formed by vectors,
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| a
= -i +
2 j + k
and b
= 3i -
2
j + 4k,
makes with the coordinate axes. |
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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
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| Angles, which the unit vector
c°
forms with coordinate axes, we find by using the formula, |
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| 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.
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| Solution:
Let first find
the vector d, |
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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, |
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| 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, |
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| Since vectors are coplanar each of them can be represented as linear combination of other two. |
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| 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. |
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| thus, all
given points lie on the same plane. |
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