What are the components of velocity? How does it differ from speed?

Velocity is a vector quantity. A vector quantity is a quantity that can be described by two numbers. The velocity of an object indicates how fast an object is moving and in what direction.

Geometrically, a vector is represented by a line segment with an arrow indicating its direction. The two numbers describing a vector are typically its the magnitude (the length of the line segment) and the angle between the vector and some other line. The magnitude of the velocity vector is called speed and it is the measure of how fast the object is moving. The angle indicates the direction of the motion.

The problem is that the magnitudes and angles are not easy to manipulate algebraically. The more convenient way to describe vectors is by breaking them up into components.

Consider a coordinate system where the x-axis is horizontal and the y-axis is vertical. If we have a three-dimensional vector, we would also need the z-axis, perpendicular to both x- and y- axes. If we draw the lines from the ends of a vector perpendicular to the x-axis, the segment obtained on the x-axis is the x-component, or the horizontal component, of the vector. Similarly, if we draw the lines from the ends of a vector perpendicular to the y-axis, the segment obtained on the y-axis is the y-component, or vertical component of the vector. Please see the attached image for the illustration.

Each vector is the vector sum of its components:

`veca = veca_x + veca_y`

The magnitude of the vector can be found, if the components are known, from the Pythagorean Theorem:

`a = |veca| = sqrt(a_x^2 + a_y^2)` 

If the given angle is the angle between the vector and x-axis, as on the attached image, then the components can be found from the magnitude and the trigonometric functions of the angle:

`a_x = acos(theta)`

`a_y = asin(theta)`

The components are useful because they make addition and subtraction of the vectors easy: each component of the sum of vectors equals the sum of the corresponding components, which is not true for magnitudes and angles. The equations of motions, which involve the velocity, acceleration and displacement vectors, are therefore easy to solve by breaking up each vector into components.