are two constant vectors.
Then Dot Product of two vectors is given by
3. Magnitude of Vector
4. Angle between two vectors
6. If slopes are given and angle between two curves is θ then tan θ =
7. If = = 0 ⇒ Vectors are orthogonal (θ = 90°)
8. If = = |a||b| ⇒ Vectors are parallel (θ = 0∘ )
If are two vectors then then cross product between two vectors is given by
= unit vector normal to both
3. Angle between two vectors is θ then sin θ =
then cross product between two vectors
can be calculated as
5. Geometrically cross product gives the area of triangle
6. If are two sides of triangle then area of the triangle is
1. Geometrically Triple Product gives the Volume of Tetrahedron
2. If Vector are coplanar vectors
Derivative of a Vector:
be a position vector where t is a scalar variable.
Vector differentiation is nothing but ordinary differentiation but only difference is position vector.
Vector Operator (∇ - Del)
is called vector operator
If ϕ (x, y, z) be a given scalar function then is called gradient.
1. Physically, gradient gives rate of change of ϕ w.r.t x, y, z separately.
2. Geometrically, it gives normal to the level surface.
Example 1: If ϕ = xyz then find the value of
4. Let ϕ(x,y,z) = c be given equation of the level surface then the outward unique normal vector is defined as
Find the value of unit normal vector
Angle between Two Surfaces
Let ϕ1(x,y,z) = C, ϕ2(x,y,z) = C be given equations of two level surfaces and angle between these two surfaces are given as θ then cos θ
The angle between two surfaces is nothing but the angle between their normal.
then they are said to be orthogonal surfaces
The angle between the two surfaces x2 + y2+ z2 = 9 and z = x2 + y2 − 3 at the point (2, −1, 2) is
Directional Derivatives of a Scalar Function
The directional derivative of a scalar function ϕ (x, y, z) in the direction of a vector
ve then it is in the opposite direction.
The Directional derivative of f(x, y, z) = x2yz + 4xz2 at (1, −2, −1)along (2i – j − 2k) is
Directional Derivative =
At (1, −2, −1) we have
Divergence of a Vector
If is a vector point function then is called Divergence of
Where are the functions of x, y, z
1. Divergence of a vector is scalar.
2. Physically Divergence measures (outflow - inflow)
3. A vector whose divergence is zero then it is said to be divergence free vector (or) solenoid vector i.e. outflow = inflow = constant
4. Geometrically, Divergence gives the rate at which the fluid entering in a rectangular parallelepiped per unit volume at the point.
Curl of a Vector
is called the curl of a vector where
1. If = 0 then it is said to be irrational vector otherwise it is said to be rotational vector.
2. Physically Curl gives the angular Velocity
3. Divergence of a curl of any vector is always zero.
is known as a vector triple product.
The values of a, b, c so that the vector,
Given, that vector V is irrotational
⇒ c = −1, a = 4, b = 2
∴ a = 4, b = 2, c = −1