dot product of two vectors
When two vectors are multiplied with each other and answer is a scalar quantity then such a product is called scalar product or dot product of vectors.
A dot (.) is placed between vectors which are multiplied with each other that’s why it is also called “dot product”.
Scalar = vector .vector
Vector dot product example
- The product of force F and displacement S is work “W”.
i.e W =F . S
- The product of force F and velocity V is power “P”.
i.e P =F . V
- The product of electric intensity E and area vector A is electric flux Φ.
i.e Φ = E . A
Dot product formula
The product of magnitudes of vectors and the cosine of angle between them.Consider two vectors A and B making an angle θ with each other.
A . B = AB Cos θ
Where “B Cos θ ” is the component of B along vector A and 0 ≤ θ ≤ π.
Dot product properties
- If vector A is parallel to B then their scalar product is maximum.
i.e A . B = AB Cos 0º=AB (1) =AB
- Scalar product of same vectors is equal to square of their magnitude.
A . A = AA Cos 0º=A² (1)=A²
- If two vectors are opposite to each other than their scalar product will be negative.
i.e A . B = AB Cos 180º=AB (-1) = -AB
- If vector A is perpendicular to B then their scalar product is minimum.
i.e A . B = AB Cos 90º=AB (0) = 0
- For unit vectors i ,j and k ,the dot product of same unit vectors is 1 and for different unit vectors is zero.
i.e i. i = j . j = k . k = 1
i. j = j . k = k . i = 0
Vector cross product
“When two vectors are multiplied with each other and the answer is also a vector quantity then such a product is called vector cross product or vector product.”
A cross (×) is placed between the vectors which are multiplied with each other that’s why it is also known as “cross product”.i.e
Vector = Vector × Vector
Vector cross product example
- The product of position vector “r ” and force “F” is Torque which is represented as “τ“.
i.e τ = r × F
- The product of angular velocity ω and radius vector “r” is tangential velocity.
i.e V t = ω × r
Cross product formula
The cross product is defined by the relation
C = A × B = AB Sinθ u
Where u is a unit vector perpendicular to both A and B.
Related topics in our site:
- Difference between dot product and cross product
- Types of vectors
- Difference between vector and scalar quantities