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Vectors in a Plane and Space |
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Vectors
and a coordinate system, Cartesian vectors |
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Vectors in
three-dimensional space in terms of Cartesian coordinates |
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Angles of
vectors in relation to coordinate axes, directional cosines - scalar
components of a vector |
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The unit vector of a vector |
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Vectors in
a three-dimensional coordinate system, examples |
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Vectors in
three-dimensional space in terms of Cartesian coordinates
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By introducing three mutually perpendicular unit
vectors, i,
j and
k, in direction of coordinate axes of the
three-dimensional coordinate system, called
standard basis vectors, every point
P(x,
y, z) of the space
determines the radius vector or the position vector,
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Similarly, a vector a
in the right diagram, which is directed from a point P1(x1,
y1, z1)
to a point P2(x2,
y2, z2)
in space, equals to sum of its vector components, axi,
ay j,
and
azk,
in the direction of the coordinate axes,
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x,
y, and
z
respectively, that is
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a represents the difference of the radius vectors, |
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so the scalar (numeric) components of the vector
a,
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Therefore, the length of the vector
a,
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Angles of vectors in relation
to coordinate axes, directional cosines - scalar components of a vector
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The scalar components of a vector and its magnitude form a right triangle in which the hypotenuse equals
the magnitude of the vector, then
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| since
b
= 90° -
a
then cosb
= sina, |
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components of a vector a, |
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If a vector
a
in 3D space forms with the coordinate axes, x,
y and z
angles,
a,
b
and g
respectively, then
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components of the vector are,
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while for the radius vector of the point
P(x,
y, z),
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| Therefore, for angles, a,
b
and g
hold the expressions,
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directional cosines of the vector
a.
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The unit vector of the vector
a,
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This condition must satisfy angles that a vector in three-dimensional space form with the coordinate axes.
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For vectors in a coordinate plane
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and since
b
= 90° -
a,
then |
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| cosb
= cos(90°
-
a)
= sina,
follows |
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the basic trigonometric identity.
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Similarly, the
unit vector of a radius vector
in
space
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determined by corresponding directional
cosines.
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Therefore the radius vector forms with coordinate axes
angles,
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As vectors are uniquely determined by its components or coordinates, they are usually denoted using
matrix algebra notation,
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in the coordinate plane,
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space |
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Vectors in
a three-dimensional coordinate system examples |
| Example:
Determine angles that a radius vector of the point A(3,
-2, 5) forms with the coordinate axes. |
| Solution:
Let calculate the magnitude or length of the radius vector, |
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| Angles between the radius vector and the coordinate axes are, |
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| Example:
A vector AB is directed from point
A(-1,
-2,
1) to point B(-2,
3, 4), find the unit vector of the
vector AB. |
| Solution:
Determine the vector AB
from the expression |
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| The length of the vector
AB |
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| The unit vector of the vector
AB |
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| Check
that the directional cosines of the unit vector satisfy the relation, |
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Example: A vector
a
in a 3D-space, of the length | a
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= 4, forms with axes,
x and
y
the same angles, |
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= b
= 60°, find the
components (coordinates) of the vector a. |
| Solution:
Using relation |
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| applying given conditions, |
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| Example: Show that vectors,
a
= -i
+ 3 j
+ k,
b
= 3i -
4 j -
2k
and c
= 5i -
10 j -
4k
are coplanar. |
| Solution:
If all three vectors lie on the same plane then there are coefficients,
l and
m such that, for example
c
= la
+ mb,
i.e., each of the vectors can be expressed as the linear combination of the remaining two. |
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| Example: Points,
A(0,
-2,
1), B(-2,
1, -3) and
C(3,
-1,
2) are the vertices of a triangle, determine the
vector of the median mc
= CM
and its length. |
| Solution:
The radius vector of the midpoint of the
side AB, |
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| The vector of the median
CM,
mc
= CM
= rm -
rc, |
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| and the length of the median
CM,
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| To check over the obtained result, calculate the coordinates of the centroid
G, |
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| The centroid divides every median in the ratio
2 : 1, counting from the vertex to the midpoint, therefore |
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