- A unit vector is the vector whose magnitude is 1 unit. It is used to specify the direction of the given vector.
- Unit vector is defined as:
- Magnitude of a vector is:
- A vector can be further broken down or resolved into its two components: vertical and horizontal. Vertical component is defined as and horizontal component is defined as .
Vector quantities are extremely useful in physics. The important characteristic of a vector quantity is that it has both a magnitude and a direction. Both of these properties must be given in order to specify a vector completely.
A unit vector is the vector whose magnitude is 1 unit. It is used to specify the direction of the given vector. Also:
If a vector is divided by its magnitude (modulus) then we get a unit vector in the direction of that vector.
Unit vectors can be described as i + j+ k, where i is the direction of the x axis and j is the direction of the y axis and k is the direction of the z axis.
If r = ai +bj+ck, then the unit vector in the direction of r is given by:
The denominator of the above formula is basically the magnitude of the vector, therefore to work out the magnitude of this vector, we use the formula:
Addition of Vectors
Given two vectors a and b, we form their sum a + b, as follows.
We translate the vector b until its tail coincides with the head of a.
Then, the directed line segment from the tail of a to the head of b is the vector a+b just as shown in Fig 1.
Q. Find the sum of vectors p = 2i + 5j +3k and q = i – 2j + 2k.
We carry out the addition of two vectors in the following way:
p + q = (2 + 1)i + (5 – 2)j + (3 + 2)k
p + q = 3i + 3j +5k
Q. Work out a – b when a = 10i + 12j – 7k and b = 5i – 2j + 8k
a – b = (10 – 5)i + (12 – (-2))j + (-7 – 8)k
a – b = 5i+ 10j- 15k
Resolving a Vector
A vector can be further broken down or resolved into its two components; vertical and horizontal.
Let’s take an example of a vector in figure 2 (Fig 2)
This vector can be resolved into two components:
1. Vertical component which is along the y axis.
2. Horizontal component which is along the x axis
We can calculate the vertical and horizontal components if we know the magnitude and direction of the vector.
In other words: we can work out the components of the vector if we know the length of the line and the angle between it and the horizontal or vertical axis.
Let’s suppose vector r in Fig 2, its vertical component is defined as , and its horizontal component is defined as .
Q. Find vector r given that magnitude of and angle between the vector and its horizontal component is 30°.
We resolve the vector into two components as shown in Fig 3 below.
- vertical component:
- horizontal component:
r = 5i + 8.66 j