Vectors Concepts Page - 4

Definition

Cross Product of Vectors

The Cross Product a×b of two vectors is another vector that is at right angles to both.
a⃗ ×b⃗ =absin(θ)n^
Where n^ is perpendicular to both the vectors. 

Definition

Linear combination of orthogonal vectors

Let us define a set S$S$ of three perpendicular vectors that can be defined in the rectangular coordinate system.
S={S⃗ 1,S⃗ 2,S⃗ 3}
Then, any vector A⃗  in the rectangular coordinate system can be represented as a linear combination of the three vectors in the set S. This is done by taking a dot product along each component.
Example:
A vector is defined as A⃗ =3i^. Rewrite A⃗ $\stackrel{}{ }$as a linear combination of S⃗ 1=2i^+j^ and S2=i^−2j^.
Unit vectors are: S1^=2i^+j^5√, S2^=i^−2j^5√
Component of A along S1 is : A1=A⃗ .S1^=6/5√
Component of A along S2 is : A2=A⃗ .S2^=3/5√
Then, A⃗ =A1S1^+A2S^2
Solving, A⃗ =65S⃗ 1+35S⃗ 2
Note:
1. Orthogonality of the given vectors is a necessary condition to write a vector as a linear combination along the vectors.
2. Number of orthogonal unit vectors is equal to the dimension of the coordinate system.

Formula

Cross Product of Vectors

a⃗ ×b⃗ =0 (For parallel vectors)
a⃗ ×b⃗ =abn^ (For perpendicular vectors)
(Where n^ is perpendicular to both the vectors)

Definition

Right hand thumb rule

Direction of the resultant of the cross product of two vectors can be found using the right hand thumb rule. It is shown in the attached figure.
For example, if a⃗  is in east direction and b⃗ $\stackrel{}{ }$is in north direction, then a⃗ ×b⃗  is in the upward direction.

Example

Dot Product of Two Vectors expressed in rectangular coordinate system

Representation of vectors in rectangular coordination system:
a⃗ =a1i^+a2j^
b⃗ =b1i^+b2j^
Their dot product is:
a⃗ .b⃗ =a1b1+a2b2

Example

Cross Product of Vectors

Representation of vectors in rectangular coordination system:
a⃗ =a1i^+a2j^
b⃗ =b1i^+b2j^
Their cross product is:
a⃗ ×b⃗ =(a1b2−a2b1)n^

Example

Dot Product of two Perpendicular Vectors

   let,
   A⃗  and B⃗  are perpendicular vectors, hence angle between them is 90o.
   using definition of scalar product we get,
   A⃗ ⋅B⃗  = AB cosθ  = AB cos(90) = 0

Example

Dot product of two parallel vectors

   let,
   A⃗  and B⃗  are parallel vectors, hence angle between them is zero.
   using definition of scalar product we get,
   A⃗ ⋅B⃗  = AB cosθ  = AB cos0 = AB

Result

Angle Between Vectors

cosθ=a⃗ .b⃗ |a||b|

Result

Component of a Vector

Since a⃗ .b⃗ =abcosθ
Projection of vector b on a is: bcosθ

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