There are two ways to represent a 2D vector, v:
1.
(v, θ)
·
v is the magnitude of the vector v.
·
θ is the angle that the vector’s
direction makes with the x-axis.
2.
v = vxi + vyj
·
vx is
the x-component of the vector.
·
vy
is the y-component of the vector.
The second representation is more
convenient for problems. An intuitive connection between the two
representations is this: The vector v can be represented by an
arrow drawn from (0, 0) to (vx, vy). In that case the
length of the arrow will be v and it will make an angle θ with the x-axis.
Problem
Solving Tips:
Tip
1.1:
Drawing vectors: Put the tail of the
arrow at the point of application – e.g., the arrow representing a force vector,
F, should have the tail on the body on which F is applied. A common
mistake (especially for force vectors) is to put the head at the point of
application.
Tip
1.2:
Going from (vx, vy)
form to (v, θ) form:
Tip
1.3:
The component of a vector v
along a certain direction is v.cos θ where θ is the included angle
between the direction of the vector and the direction of the component.
This is the “shadow = cos θ” tip. (shadow = the
component of the vector with sun at noon.)
The component perpendicular to the above
is v.cos (90-θ) = v.sin θ.
Tip
1.4:
The component of a vector along a
direction perpendicular to itself is 0. This principle of orthogonality is used
to simplify problems.
Tip 1.5:
Up-down intuition:
As the angle θ that the vector makes with
the horizontal increases from
to
, the horizontal component vx decreases.
The arrows in the notation represent this relationship. The relationship vx
= v.cos θ is not exactly linear.
θ
↑ vx ↓
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