# Vectors

## Vector Acceleration and Speed

### Position Vector

Imagine a piece of furniture moving on a random path with an O origin.

If we place a Cartesian plane situated on this origin, then we can locate the furniture in this trajectory by means of a vector.

The vector it is called displacement vector and has modulus, direction and direction.  = P-O

### Vector Speed

Vector Average Speed: Consider a moving along the trajectory of the chart above, occupying positions and in moments and respectively.

Knowing that the average velocity is equal to the time-shift vector quotient:  Note:

The average velocity vector has the same direction and direction as the displacement vector because it is obtained when we multiply a positive number. by the vector .

Instantaneous Speed ​​Vector: Analogous to instantaneous scalar speed, when the time interval tends to zero ( ), the calculated speed will be the instantaneous speed.

So: ### Vector Acceleration

Vector Average Acceleration: Considering a piece of furniture that travels any trajectory with speed in an instant and speed at a later time , its average acceleration will be given by:  Note:

As for the velocity vector, the acceleration vector will have the same direction and direction as the velocity vector, as it is the result of the product of this vector ( ) by a positive scalar number, .

Instant Acceleration Vector: Instant vector acceleration will be given when the time interval tends to zero ( ). Knowing these concepts, we can define the functions of velocity as a function of time, displacement as a function of time, and the Torricelli equation for vector notation: For example:

A body moves with speed , and constant acceleration , as described below: (a) What is the velocity vector after 10 seconds? (b) What is the position of the furniture at this moment?

(a) To calculate vector velocity as a function of time, we need to decompose the initial velocity and acceleration vectors into their projections into x and y: So we can divide the motion into vertical (y) and horizontal (x):

In x:  In y:  From these values ​​we can calculate the velocity vector:  (B)Knowing the velocity vector, we can calculate the position vector by the Torricelli equation, or by the hourly displacement function, both in the form of vectors: By Torricelli: in the same direction and direction as the acceleration and velocity vectors.

By Time Position Function: in the same direction and direction as the acceleration and velocity vectors.