Recall
A vector is a quantity that has both magnitude and direction
• Adding vectors
• subtracting vectors
Recall
• Equal vectors (~a = ~b): same magnitude (|~a| = |~b|) and same direction.
• Opposite vector (~a = −~b): same magnitude (|~a| = |~b|) and opposite
direction.
• Parallel vectors (~a k ~b): same or opposite direction
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Multiplying a Vector by a Scalar
If we take a vector ~v and multiply it by a scalar k (any real number), we are
performing scalar multiplication and have produced the scalar multiple k~v .
The vector k~v will be |k| times as long as ~v and it will be parallel to ~v .
The magnitude of k~v is |k~v | = |k||~v |.
If k > 0, k~v is in the same direction as ~v .
If k < 0, k~v is in the opposite direction to ~v .
If |k| > 1, k~v is longer than ~v .
If |k| < 1, k~v is shorter than ~v .
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The properties of scalar multiplication:
For any vectors ~u and ~v and scalars k and c,
(i) k(~u + ~v ) = k~u + k~v (distributivity)
(ii) k(c~v ) = (kc)~v (associativity)
(iii) 1~v = ~v (identity)
Example:
Simplify 2(3~u − ~v ) + 4~v .
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Vectors that are parallel are also said to be collinear because they would lie
on a straight line when arranged tail to head. Also, they are scalar multiples
of each other.
i.e. there is some k ∈ R such that ~u = k~v .
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Linear Combination:
A vector of the form s~u+t~v (where s and t are scalars) is called a linear combination
of vectors ~u and ~v .
Example:
~ k DC
~ and DC = 4AB. Write DC,
~ AC
~ and
In the trapezoid ABCD, AB
~ as a linear combination of DA
~ and AB.
~
BC
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Applications of Vector Addition
Example:
A cannonball of mass 100 Kg is fired horizontally out of a cannon with
a force of 2500 N. Gravity will act vertically (downward) with a force of
(9.8 m/s2 )(100 Kg) = 980 Kgm/s2 = 980 N. Find the resultant force F~ .
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Rectangular Vector Components
Two vectors that are perpendicular to each other and add together to give a
vector ~v are called rectangular vector components of ~v .
Any vector can be resolved into rectangular (perpendicular) components,
typically horizontal and vertical.
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Example:
A child pulls a wagon with a force of 40 N at an angle of 25◦ to the horizontal.
Find the horizontal and vertical component of the force.
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Example:
An object that weighs 75 N is resting on an inclined plane that makes an
angle of 15◦ with the horizontal. Draw all the forces acting on the object
and find their vertical and horizontal components.
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~ is a vector that balances another vector or combiAn equilibrant vector E
nation of vectors and hence is equal in magnitude and opposite in direction
~
to the resultant R.
Tension is the equilibrant force in a rope or chain that keeps an object in
place (or stationary).
Example:
A picture that weighs 60 N is hanging from a wire (attached to the picture
frame) placed on a hook on the wall such that the hook is in the centre of
the wire and the two segments of wire have an angle of 120◦ between them.
Find the tension on each segment of the wire.
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