- A quantity that has magnitude as well as direction is called vector.From a geometric point of view, a vector can be defined as a line segment having a specific direction and a specific length
- It is denoted by the letter bold letter a or it can be denoted as a
- Magnitude of a vector a is denoted by |a| or a.It is a positive quantity

Physical quantities may be divided into two categories

(1) Scalars are physical quantities that only have magnitude for example mass, length, time, temperature etc.

(2)Vectors are physical quantities having both magnitude and direction for example velocity, force, electric field, torque etc. It can be represented by an arrow in space.

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(iX)

(1) Vector addition is commutative i.e.

(2) Vector addition is associative i.e.

(3)

4)

Now

So we will first reverse the direction of vector

From the terminal point of

Another method to find substraction of vectors would be

Let draw vector

Scalar multiplication of vectors is distributive i.e.,

n(

(i) |k a| = |k| |a|

(ii) k O = O

(iii) m (-a) = – ma = – (m a)

(iv) (-m) (-a) = m a

(v) m (n a) = mn a = n(m a)

(vi) (m + n)a = m a+ n a

(vii) m (a+b) = m a + m b

Then we can represent any vector in the components forms like

r= x

- x,y and z are scalar components of vector r
- x
**i**,y**j**,z**k**are called the vector components - x,y,z are termed as rectagular components
- Length of vector or magnitude of the vector is defined as
- x,y,z are called the direction ratio of vector r
- In case it is given l,m,n are direction cosines of a vector then

In component form addition of two vectors is

Where,

Thus in component form resultant vector C becomes,

Cx = Ax+ Bx : Cy = Ay+ By : Cz = Az+ Bz

In component form substraction of two vectors is

D = (Ax- Bx)

where, A = (Ax, Ay, Az) and B = (Bx, By, Bz)

Thus in component form resultant vector D becomes,

Dx = Ax - Bx : Dy = Ay- By : Dz = Az- Bz

Equality of vector in components form

Axi +Ayj+Azk= Bxi+Byj+Bzk

Ax=Bx

Ay=By

Az=Bz

Multiplication of scalar by vector in components form

=K(Bx

=(kBx)

In the coordinate system, the line joining the origin O to the point P in the system is called the position vector of Point P

Let Point P and Q are there

Postion Vector of Point P

Postion Vector of Point Q

=(x

Vectors a and b are collinear, if a = λb, for some non-zero scalar λ.

Let A, B, C be any three points.

Points A, B, C are collinear <=> AB, BC are collinear vectors.

<=> AB = kBC for some non-zero scalar k

Let A and B be two points with position vectors

(i) Let P be a point dividing AB internally in the ratio m : n. Then,

where r is the position vector of point P

(ii) The position vector of the mid-point of

(iii) Let P be a point dividing AB externally in the ratio m : n. Then,

- Dot product of two vectors
**A**and**B**is defined as the product of the magnitudes of vectors A and B and the cosine of the angle between them when both te vectors are placed tail to tail. Dot product is represented as**A**.**B**thus,

where θ is the angle between two vectors and 0 =< θ <= 180

- Result of dot product of two vectors is a scalar quantity.
- Dot product is commutative :
**A**.**B**=**B**.**A** - Dot product is distributive :
**A**. (**B**+**C**) =**A.B**+**A**.**C**also**A**.**A**= |**A**|2 - In component form dot product of two vectors

Thus for calculating the dot product of two vectors, first multiply like components, and then add.

- For two mutually perpendicular vectors
**A**.**B**= 0 - Scalar product of two equal vectors is
**A**.**A**=|A|_{2} - If either a or b is the null vector, then scalar product of the vector is zero.
- If a and b are two unit vectors, then a * b = cos θ

and

Angle between Two Vectors If θ is angle between two non-zero vectors, a, b, then we have

a * b = |a| |b| cos θ

cos θ = a * b / |a| |b|

- Cross product or vector product of two vectors
**A**and**B**is defined as

where

- Cross product is distributive i.e.,
**A x (B + C) = (A x B) + (A x C)**but not commutative and - The cross product of two parallel vectors is zero.
- Component of any vector is the projection of that vector along the three coordinate axes X, Y, Z.

= (AyBz - AzBy)

- Cross product of two vectors is itself a vector.
- To calculate the cross product, form the determinant whose first row is x, y, z, whose second row is A (in component form), and whose third row is B.

- For three vectors A, B, and C, their scalar triple product is defined as
**A . (B x C) = B . (C x A) = C . (A x B)**obtained in cyclic permutation.

- Switching the two vectors in the cross product negates the triple product, i.e.:
**C . (A x B) =-C . (B x A)** - If
**A**= (A_{x}, A_{y}, A_{z}) ,**B**= (B_{x}, B_{y}, B_{z}) , and**C**= (C_{x}, C_{y}, C_{z}) then,**A . (B x C)**is the volume of a parallelepiped having A, B, and C as edges and can easily obtained by finding the determinant of the 3 x 3 matrix formed by**A**,**B**, and**C**.

- If the scalar triple product is equal to zero, then the three vectors a, b, and c are coplanar, since the "parallelepiped" defined by them would be flat and have no volume.

- For vectors A, B, and C, we define the vector tiple product as
**A x (B x C) = B(A . C) - C(A - B)**

- Note that (A × B) ×C ≠ A× (B × C)

- What is vector
- |
- Difference between Scalar and Vector
- |
- Type Of Vectors
- |
- Addition Of vector
- |
- Subtraction of vectors
- |
- Scalar multiplication of vectors
- |
- Components of the vector
- |
- Multiplication of two vectors
- |
- Dot product of vectors
- |
- Cross product of vectors
- |
- Triple product
- How to solve vector algebra problems

Class 12 Maths Class 12 Physics Class 12 Chemistry Class 12 Biology

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