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Vectors in a Plane and Space |
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Vectors in
three-dimensional space in terms of Cartesian coordinates |
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Vectors in
a three-dimensional coordinate system, examples |
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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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| College
algebra contents
F |
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| Copyright
© 2004 - 2020, Nabla Ltd. All rights reserved. |
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