Tuesday, November 11, 2014

Revision Note 1: Vectors and components of vectors

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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