Part 13 : Vector Components

Any two dimensional vector could be broken down into components (which aren’t similar to elements of vector). These are the horizontal and vertical components of vector.

Vertical and Horizontal component of Vector V represented with orange and blue respectively

The numeric value of horizontal component is the product of magnitude of vector with cosine of theta(θ) = |V|cos(θ)

and vertical component is product of magnitude of vector with sine of theta(θ) = |V|sin(θ) .

Where theta(θ) is the angle of that vector with x-axis in anticlockwise direction.

Breaking vectors such as force applied at a particular angle into its vector components helps us solve problems in classical mechanics.

These numeric values of components are derived from the unit circle definition of trigonometric ratios. Which could be explained as :

If we draw a circle on Cartesian plane having center at origin (0,0) with radius = 1 unit, then the line drawn from center to the edge of circle will intersect at

(x,y) = cos(θ), sin(θ)

where theta(θ) is the angle between that line and x-axis in anticlockwise direction. This is represented below

Unit Circle definition of cosine and sine

Now, we will draw a circle having center at origin with radius equal to magnitude of vector V.

Circle which has center at origin and radius = |V|

The point at which vector V intersects the circle will be

(x,y) = |V|cos(θ), |V|sin(θ)

Hence, any two dimensional vector V could be broken down into horizontal and vertical components.

Additional Resources
The unit circle definition of sine, cosine, and tangent by Khan Academy

Vector components from magnitude & direction by Khan Academy

Classical Mechanics

Read Part 14 : Dot and Hadamard Product

You can view the complete series here
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