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Vectors in a Plane and Space |
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Vectors
and a coordinate system, Cartesian vectors |
Angle between vectors
in a coordinate plane |
Projection of a vector in the direction of another vector,
the scalar and vector components |
Vectors examples |
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Angle between vectors
in a coordinate plane |
From the scalar product
a
· b = |
a |
· | b | · cosj
derived are, the angle between two vectors, formulas |
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- for plane vectors |
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- for space vectors |
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The cosine of the angle between two vectors,
a
and b
represents the
scalar product of their unit
vectors, |
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Projection of a vector in the direction of another vector,
the scalar and vector components |
The scalar
component |
The length of projection of
a
in the direction of
b
or the scalar component ab,
from the diagram, |
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Thus, the scalar component of a vector
a
in the direction of a vector b
equals the scalar product of the vector
a
and the unit vector b0
of the vector
b.
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The vector component |
By multiplying the scalar component
ab, of a vector
a in the direction of
b, by the unit vector of the vector
b0
of the vector
b, obtained
is |
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or |
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Vectors
examples |
Example:
Applying the scalar product, prove
Thales’ theorem which states that an angle inscribed in a |
semicircle is a right angle. |
Solution:
According to diagram in
the right figure |
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since square of a vector equal to square of its length, |
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thus,
a
and b are orthogonal vectors as |
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Example:
Prove the law of cosines used in the trigonometry of oblique triangles. |
Solution:
Assuming the directions of vectors as in
the right diagram |
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Using the scalar product and substituting square of vectors
by square of their lengths, obtained is |
a2 =
b2 + c2 -
2bc · cosa
- the law of cosines. |
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In the same
way, |
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Example:
Determine a parameter l so the given vectors,
a
= -2i
+ l
j
-
4k
and b
= i -
6 j + 3k
to be perpendicular. |
Solution:
Two vectors are perpendicular if their scalar product is zero, therefore |
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Example:
Find the scalar product of vectors, a
= -3m
+ n
and b
= 2m -
4n if
| m |
= 3 and
| n |
= 5 , and the
angle between vectors, m
and n is
60°. |
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Example:
Given are vertices, A(-2,
0, 5), B(-3, -3,
2), C(1,
-2,
0) and
D(2,
1, 3), of a parallelogram, |
find the
angle subtended by its diagonals as is shown in the diagram below. |
Solution:
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Example:
Given are points, A(-2,
-3,
1), B(3,
-1,
-4), C(0,
2, -1) and
D(-3,
0, 2), determine the scalar and
vector components of the vector AC
onto vector BD. |
Solution:
The scalar component of the vector
AC
onto vector BD, |
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The vector component of the vector
AC
in the direction of the vector BD
equals the product of the scalar component ACBD
and the unit vector BD°
that is
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