As we know Vector is a quantity which has some magnitude and direction. Any Vector can be split into two Vector components such that the two Vector component are perpendicular to each other.
For example as shown in the figure given below a Vector in the north west direction can be split into two components which are in north and west direction. Similarly a Vector in north east direction can be divided into two components which are in north and east direction. Note that the magnitude of these two components will not be same as that of the initial Vector.
Vector Components and the Vector which is divided into components together form a right angled triangle such that the two components are base and altitude of the triangle and original Vector if hypotenuse of the triangle. Given below a Vector ‘a’ which is divided into two components ax and ay such that ax, ay, a form the base height and hypotenuse of the triangle. ay component shown in figure can be shifted in right direction parallel to form the height of the triangle.
Components of a Vector can be found out by using properties of trigonometry which are based on a right angled triangle.
According to trigonometry, in a right triangle if hypotenuse makes an angle Ө with the base then,
sinӨ = height or perpendicular/hypotenuse
cosӨ = base/hypotenuse.
For previous illustration sinӨ = ay/a
cosӨ = ax/a
So, Vector Component ax and ay will be:
ax = a(cosӨ) ……….(component of Vector ‘a’ in x direction)
ay = a(sinӨ) ………….(component of Vector ‘a’ in y direction)
Vector a can be written as: a = axi + ayj = a (cosӨ)i + a (sinӨ) j
If a Vector is given as a = a1i + a1j, then the Vector components can be read directly. Here a1 is component in x (horizontal) direction and a2 is Vector in y (vector) direction.
Let us take an example of Components of Vectors :
Example) Find the component of a Vector with magnitude 5 and makes an angle of positive 60o with horizontal.
Solution) let the Vector be P.
components of Vector P in x direction will be: Px= PcosӨ
= 5cos(60o)
= 5(1/2)
= 5/2 = 2.5
components of Vector P in y direction: Py= P(sin60o)= 5(√3/2)
So Vector P = 2.5i +5(√3/2)j
Note) If the Vector makes 60o with the vertical then Py = Pcos(60o) and Px = Psin(60o) or Py = Psin(30o) and Px=Pcos(30o) (This is because in this case the Vector will make 30o with horizontal)
For example as shown in the figure given below a Vector in the north west direction can be split into two components which are in north and west direction. Similarly a Vector in north east direction can be divided into two components which are in north and east direction. Note that the magnitude of these two components will not be same as that of the initial Vector.
Vector Components and the Vector which is divided into components together form a right angled triangle such that the two components are base and altitude of the triangle and original Vector if hypotenuse of the triangle. Given below a Vector ‘a’ which is divided into two components ax and ay such that ax, ay, a form the base height and hypotenuse of the triangle. ay component shown in figure can be shifted in right direction parallel to form the height of the triangle.
Components of a Vector can be found out by using properties of trigonometry which are based on a right angled triangle.
According to trigonometry, in a right triangle if hypotenuse makes an angle Ө with the base then,
sinӨ = height or perpendicular/hypotenuse
cosӨ = base/hypotenuse.
For previous illustration sinӨ = ay/a
cosӨ = ax/a
So, Vector Component ax and ay will be:
ax = a(cosӨ) ……….(component of Vector ‘a’ in x direction)
ay = a(sinӨ) ………….(component of Vector ‘a’ in y direction)
Vector a can be written as: a = axi + ayj = a (cosӨ)i + a (sinӨ) j
If a Vector is given as a = a1i + a1j, then the Vector components can be read directly. Here a1 is component in x (horizontal) direction and a2 is Vector in y (vector) direction.
Let us take an example of Components of Vectors :
Example) Find the component of a Vector with magnitude 5 and makes an angle of positive 60o with horizontal.
Solution) let the Vector be P.
components of Vector P in x direction will be: Px= PcosӨ
= 5cos(60o)
= 5(1/2)
= 5/2 = 2.5
components of Vector P in y direction: Py= P(sin60o)= 5(√3/2)
So Vector P = 2.5i +5(√3/2)j
Note) If the Vector makes 60o with the vertical then Py = Pcos(60o) and Px = Psin(60o) or Py = Psin(30o) and Px=Pcos(30o) (This is because in this case the Vector will make 30o with horizontal)
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