- Unit vector is defined as:
- Position vector of two points is:
- Equation of a vector line is where a is the point on the line and b is the direction
- Angle between two vectors is
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.
Scaler & Vector
A scalar quantity is the one which is completely represented by its magnitude, in terms of a given unit i.e mass (kg), energy (joules) etc are examples of a scalar quantity.
Whereas a vector quantity is the one which is completely represented by its magnitude in terms of a given unit and its direction from point of reference i.e acceleration, displacement, velocity etc.
Let r be a given vector. Using the above definition of a vector quantity, we can write vector r as:
Where is the magnitude of the vector r and is the unit vector in the direction of r.
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, where i is the direction of the x axis and j is the direction of the y axis.
If r = p i +q j, then the unit vector in the direction of r is given by:
If ”O” is the origin then the position of any point A can be determined by the vector which is called the position vector of A with respect to the origin O.
We can say that if a is the position vector of the point A, then:
Let a and b be two position vectors of the points A and B with respect to the origin O respectively as shown in Fig 1.
Similarly, we can express other vectors in terms of their position vectors i.e:
Magnitude of a vector
For example for a given vector a i + b j, we can find its magnitude using the formula:
Q. Find the magnitude of the vector 5 i + 4 j
Vector Equation of a line
Let’s suppose a line is parallel to a direction vector b and passes through fixed point A with position vector a. Let r be the position vector of a general point on the line.
Thus the equation of the line is defined by where is a parameter and corresponds to a point on the line.
Q. Find the equation of the line through the points (2, -3) in the direction 3 i – 5 j
We know the point a = (2,-3) and b = (3, -5)
Substitute value in the above equation we get:
When the product of two vectors is scalar, it is called a scalar product or dot product.
Remember these points about the scalar product;
1. Scalar product is commutative:
2. Scalar product is distributive: where a, b, c are three vectors
Suppose we are given two vectors:
Then the scalar product of p and q would be:
Angle between two vectors
If is the angle between two vectors a and b, then:
or can be written as
Note: When the dot product is equal to zero, it means two vectors are perpendicular to each other as cos 90°=0