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Vectors in a Plane and Space |
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Vectors
in a Plane |
Vectors - introduction |
Length, magnitude or
norm of the vector |
Collinear, opposite
and coplanar vectors |
Addition of vectors |
Triangle rule (law)
and parallelogram rule |
Zero or null vector |
Addition, subtraction
and scalar multiplication of vectors, examples |
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Vectors - introduction |
There are physical quantities like force, velocity, acceleration and others that are not fully determined by their
numerical data.
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For example, a numerical value of speed of motion, or electric or magnetic field strength, not
give us the information about direction it move or direction they act.
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Such quantities, which are completely specified by a magnitude and a direction, are called vectors or vector
quantities and are represented by directed line segment. |
Thus,
a vector is denoted as
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where the point
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A is called the tail or start and point
B, the head or
tip.
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The length or magnitude or norm of the vector
a
or |
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is |
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Therefore, the length of the arrow represents the vector's magnitude, while
the direction in which the arrow points, represents the vector's direction. |
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A vector with no magnitude, i.e., if the tail and the head coincide, is called the
zero or null vector
denoted |
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Collinear, opposite
and coplanar vectors |
Two vectors are said to be
equal if they have the same magnitude
and direction or if by parallel shift or translation one could be
brought into coincidence with the other, tail to tail and head to
head.
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Vectors are said to be
collinear if they lye on the same line or on
parallel lines.
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Vectors,
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in
the above figure are
collinear. |
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Two collinear vectors of the same magnitudes but opposite directions are said to be
opposite vectors. |
A vector that is opposite to |
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is
denoted |
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as shows the
above right figure. |
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Three or more vectors are said to be
coplanar if they lie on the same plane. If two of three vectors are
collinear then these vectors are coplanar.
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To prove this statement, take vectors,
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of which
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are
collinear.
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By using translation bring the tails of all three vectors at the same point. Then, the common line of
vectors,
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and the line in which lies the vector
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determine the unique plane. |
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Therefore, if vectors are
parallel to a given plane, then they are coplanar.
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Addition of vectors |
The sum of vectors, |
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can be obtained graphically by placing the tail of |
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to the tip or head of |
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using translation.
Then, draw an arrow from the initial point (tail) of |
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to the endpoint
(tip) of |
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to obtain |
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the
result. |
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The
parallelogram in the above figure shows the addition |
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where, to the tip of |
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by translation, placed |
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is the tail of |
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then, drawn
is the resultant |
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by joining the tail of |
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to the tip of |
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Note that the tips of the resultant and the second summand should coincide. |
Thus, in the above figure
shown is, the triangle
rule (law) and the parallelogram
rule for finding the
resultant or |
the addition of the two given vectors. The result is the same
vector, that is
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Therefore, vector addition is commutative. |
Since vectors, |
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form a triangle, they lie on the same
plane, meaning they are coplanar. |
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Addition of three vectors,
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is defined as
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and represented graphically
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The
above diagrams show that vector addition is
associative, that is |
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The same way defined is the sum of four vectors. |
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If by adding vectors obtained is a closed polygon, |
then the sum is a null vector. |
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By adding a vector |
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to its opposite vector |
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graphically it leads back to the initial point, therefore |
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so, the
result is the null vector. |
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Addition, subtraction
and scalar multiplication of vectors, examples |
Example: |
Given are vectors,
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determine
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Example:
Given is a regular hexagon
ABCDEF with the center
O. Express vectors
CD,
BE,
EA,
and CE,
in terms of vectors, AB
= a
and BC
= b.
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Solution:
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Example: Determine the distance of the midpoint
M, of the segment
P1P2, and the point
O, if points,
P1
and P2
are heads of vectors p1
and p2 respectively, and whose tails coincide
with the point O
as shows the figure. |
Solution: The
vector P1P2
represents the difference p2
-
p1. |
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Example: In a triangle
ABC drown are medians as vectors,
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Prove
that
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Solution:
Replacing the sides of the triangle by vectors, directed as in
the diagram,
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Example:
Use vectors to prove that line segments joining the midpoints of adjacent sides of a quadrangle,
form a parallelogram.
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Solution: The sides of the quadrangle ABCD are replaced by vectors, directed as in
the diagram so that the opposite vectors connecting midpoints are, |
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As equal vectors are of same lengths and parallel, therefore the line segments connecting the midpoints of any quadrangle, form a parallelogram. |
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Example:
Determine a vector which coincides with the angle bisectors of vectors, a
and b
in the diagram.
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Solution: The unit vectors of the given vectors
are, |
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and they form a rhombus whose diagonal is |
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That is, a vector |
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will coincide
with |
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the angle bisector, while a vector |
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defines
the angle bisector of the supplementary angle of vectors, a
and b.
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