How to define vector, what's a vector.
A vector is a quantity that has magnitute and detection types of vector commonly used in physics course are displacement vector velocity vectors and force vector.
All other types vectors are treated in the same as displacement vectors, a vector is represented as a line with a point on it just like an arrow.
The length of the Lines represent the magnitude of the quantity the direction of the arrow (line) represent the direction of the displacement it doesn't matter what length we use to represent the imagnitude of a vector but it's important that the lengths of all the vector used in one vector diagram have the right proportion in relation to each other in other words if we decide to represent a displacement of negritude 5m with a line belong that a displacement of magnitude 10mls (twice as long) most be represented with a line 12cm long.
Addition of vector
We often need to combine vector we can add or subtract vector according to a simple set of roles
In the simplest case vector are in a straight line that is in one dimension in such cases we add or subtract vectors according to the rules of arithmetic because the vectors are all in the same straight line we need only to deal with the magnitude of the quantities
exp:-
A boy start from point (A ) and walk at 3km in a straight line to point (B). Will decide to use (T) as meaning to the right any displacement to the will be designed with the negative sign (-)he then walk a further 2km to the right to point (c) then be mouse 7km from point (c)to point (x)
1). What's his Ouran displacement
2). What distance has he walked
Solution
Let's draw the situation using arrows to represent the displacement vectors all are on a straight line noted earlier.
Below is a photo of the Drawing 👇👇👇
 |
In the diagram above will have used two parallel lines simplyto show the situation more clearly.
It's over all displacement it's form A to x where it stopped their magnitude of the displacement is starting (A), T 3km, T 2km =5km to point c now be Walk left -7km to point x. The total overall displacement is therefore 2km from starting point.
The distance he has walked is a scalar quantity we simply add the distance direction left or right is unimportant because scalars have no direction clearly the distance is 3km + 2km +7km =12km.
Vector in two dimensions
This is an example of working with displacement vectors in two dimensions
A girl start from point 0 and walk 4km in another direction she then walk 5km east of this point and arrived at a point we will call x this displacement vector is know as vector (B), draw a vector diagram to show this displacement and used the diagram to find the displacement of point x from her starting point owe start by drawing the two displacement vectors
Line A and B notice that they are arranged at right angle to each other because their direction are north and east are they have the correct relative ( proportionate) length we now draw his as another straight line the displacement vector joining o and x we call this displacement vectors.
There are in general two method of adding or compounding vectors to fine the resultant these are
I). The parallelogram method
Ii). The triangle method
The parallelogram method
The resultant of two vectors inclined to each other may be represented by the diagonal of a pallelogram draw with the two vectors as a adjacent sides in the method the two vectors are draw from a common origin and a pallelogram is constructed using these vectors as adjacent sides.
The image of figure below is the diagonal draw from the common origin
The parallelogram low of vectors status this is if two vectors are represented in magnitude and directed by the adjacent sid of a pallelogram the diagram of the pallelogram drawn for the point of intersection of the vectors represent the reactant vectors in magnitude and direction
Exp :- fine the resucent of two vectors of 3 unit and 4 unit acting of a point out an angie of 45° with each other
Solution
OC² =OA² + OB²–(OA)×(OB)×005 135°
=OA² + OB²+2(0A)×(oB)Co8(180-135)°
=3²+4²+2×3×4 COS 45°
=9+10+24 COS 45°
=25+24×0.7071
=41.9704
•:oc. =®41.9704
OC =6.48 unit
Using sine iules
Sin9 sin 135 0.7071
------ = ---------- = --------
Be oc oc
Sin9 = 3×0 . 7071 = 0.324
---------------
6. 45
2= sin - 0.3274
O< = 19-11⁰
The direction of the resultant vector is = 19
In general if two vectors p and a are lined at 0° to each other the resultant vector R is given by R² = p²+Q²+2p⁰ COS ø°.
The triangle method
Suppose we want to find the resultant of two vectors p and Q inclined to An Angle ø= 60⁰ to each other, the step of obtaining this resultant using the triangle method is all follows
1). Starting from a point o draw oA to represent p
2). Next draw the second vector to a scale planing it's tail at the tip of the first vector p ensuring it's magnitude and direction are correct
3). simply draw or to complete the triangle
The resultant vector in magnitude and direction.
Resolution of vector
The vector can be combined or composed to obtain a resultant vector as arready discussed in the proceeding section this resultant vector will have the some effect as the two vectors acting together at this point the reverse possible through a process known as resultant is vector.
Consider first a vector with lies in an y plane this vector can be expressed as they sum of two other vector called the components of the original vector in a given direction it's effective value in the direction for exp
The horizontal component of a vector it's effective value in a horizonted direction the process of finding component of a vector is known as the resolution of vector into it component it's costuming and most useful to resolve a vector into H component. It's Costco Mary and most useful to resolve a vector into component along morally pretending direction usually the horizontal or o< direction
Suppose a vector v is inclined at an angle ø to a horizontal direction. The horizontal component of the vector is giving by vx= v cost.
The vertical component of v is given by vy =v sin ∅
Vy and vx are at right angle to each other
EXP.
Resolve a force of 100m incline at 60⁰ to the horizontal into the horizontal vertical component
Solution
Horizontal component fx= 100 cos 50= 62.28m
Vertical component fy=100 sin 50⁰ = 25.6m
Exp A
A plane slice with a velocity of 1000km 11km in a direction m 60⁰+ find it effective velocity in the eastery and northern direction.
Velocity in the eastern direction is given vx= 1000 cos 30⁰ = 866km VR
Velocity is the northern direction is given by vy =100⁰ sin 30⁰ vy = 500km
Comments