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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 |
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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
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magnitude
and direction or if by parallel shift or
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translation one could be
brought into coincidence with
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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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