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**Dot Product**

are two constant vectors.

Then Dot Product of two vectors is given by

**Note: **

3. Magnitude of Vector

4. Angle between two vectors

5. Also,

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^{∘} )

**Cross Product **

If are two vectors then then cross product between two vectors is given by

= unit vector normal to both

**Note:**

3. Angle between two vectors is θ then sin θ =

4.

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

**Triple Product**

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.

**Formulae:**

**Vector Operator (∇ - Del)**

is called vector operator

**Gradient **

If ϕ (x, y, z) be a given scalar function then is called gradient.

**Note:**

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

**Note:**

4. Let ϕ(x,y,z) = c be given equation of the level surface then the outward unique normal vector is defined as

**Example 2:**

Find the value of unit normal vector

**Solution:**

**Angle between Two Surfaces**

Let ϕ_{1}(x,y,z) = C, ϕ_{2}(x,y,z) = C be given equations of two level surfaces and angle between these two surfaces are given as θ then cos θ

**Note:**

The angle between two surfaces is nothing but the angle between their normal.

then they are said to be orthogonal surfaces

**Example 3: **

The angle between the two surfaces x^{2} + y^{2}+ z^{2} = 9 and z = x^{2} + y^{2} − 3 at the point (2, −1, 2) is

**Solution:**

**Directional Derivatives of a Scalar Function **

The directional derivative of a scalar function ϕ (x, y, z) in the direction of a vector

given as

ve then it is in the opposite direction.

**Example 4:**

The Directional derivative of f(x, y, z) = x^{2}yz + 4xz^{2} at (1, −2, −1)along (2i – j − 2k) is**Solution:**

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

**Note:**

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

**Note:**

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.

**Example 5:**

The values of a, b, c so that the vector,

is irrotational

**Solution: **

Given, that vector V is irrotational

⇒ c = −1, a = 4, b = 2

∴ a = 4, b = 2, c = −1

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