photo.jpg

# Scalars And Vectors ## Properties Of A Null Vector - It has zero magnitude - It has arbitrary direction - It is represented by a point - When a null vector is added or subtracted from a given vector, the resultant vector is the same as the given vector - Dot product of a null vector with any othe...

# Scalars And Vectors ## Properties Of A Null Vector - It has zero magnitude - It has arbitrary direction - It is represented by a point - When a null vector is added or subtracted from a given vector, the resultant vector is the same as the given vector - Dot product of a null vector with any other vector is always zero - Cross product of a null vector with any other vector is also a null vector ## Equal Vectors - Two vectors are said to be equal if they possess the same magnitude and direction. - They can be represented by two parallel straight lines of equal length and pointing in the same direction - In the image, two equal vectors, u and v are shown. Here u = v. ## Other Types Of Vectors - **Parallel Vectors (θ = 0°)** - Two vectors (which may have different magnitudes) acting along the same directions are called parallel vectors. - In the example, vectors u and v are parallel vectors with an angle of 0° between them - **Anti-parallel Vectors (θ = 180°)** - Two vectors which are directed in opposite directions are called anti-parallel vectors. - vectors u and v shown in the image are anti-parallel, with an angle of 180° between them. - **Co-planer Vectors** - Vectors situated in one plane, irrespective of their directions, are known as co-planar vectors. - vectors P, Q and R are drawn in the x-y plane in the image. - **Negative Vector** - A vector is said to be a negative vector of another one, if it is represented by a line having the same length as that of the second and is directed in the opposite direction. - In Figure 2(b).3(ii), a vector u is shown which is the negative vector if that shown in Figure 2(b).3(i) - A negative vector can be obtained simply by putting a negative sign before a vector. - If you are having a negative vector, you can simply move it to the other side of the equation and eliminate the negative sign. - **Co-initial Vectors** - A number of vectors having a common initial point are called co-initial vectors. - Vectors u, v, w, x and y all meeting at point p are co-initial vectors. - **Collinear Vectors** - Vectors having a common line of action are called collinear vectors. There are two types of collinear vectors. - In the image, the two types of collinear vectors are shown: - **(i) Co-initial Vectors** - **(ii) Free or Non-localised Vectors** - **Localised Vectors** - Vector whose initial point (tail) is fixed is said to be a localised or a fixed vector. - Position vector of every point starts from origin. - Therefore, it can be considered as a localised or fixed vector. - **Non-localised Vectors** - Vector whose initial point (tail) is not fixed is said to be a non-localised vector or a free vector. - Vectors representing force, momentum, impulse, etc. are non-localised vectors. # Classification Of Vectors - **Polar Vectors or Radial Vectors** - Vectors whose directions are independent of the frame of reference are known as polar vectors or radial vectors. - **Free Vectors** - Vectors representing displacement, velocity, force, momentum etc. belong to this category. They are further classified into following three types: - **(i) Linear Vectors** - **(ii) Surface Vectors** - **(iii) Volume Vectors**