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Zero vector or null vector

A vector whose initial and terminal points are coincident is called the zero vector or the null vector. The magnitude of the zero vector is zero and it can have any arbitrary direction and any line as its line of support.

Unit vector

A vector of unit magnitude is called a unit vector. Unit vectors are denoted by small letters with a cap on them. If the vector 79078.png is divided by magnitude 79088.png, then we get a unit vector in the direction of 79094.png.

Like and unlike vectors

Two parallel vectors having the same direction are called like vectors.
Two parallel vectors having opposite directions are called unlike vectors.

Collinear vectors

Vectors which are parallel to the same vector and have either initial or terminal point in common are called collinear vectors. If vectors 79102.png and 79108.png are collinear then 79115.png, where λ is scalar. Collinear vectors are called dependent vectors.


  • Two non-zero vectors 82673.png and 82669.png are collinear iff there exists scalars xy not both zero such that 82665.png
  • If 82659.png are any two non-zero non-collinear vectors and xy are scalars, then 82654.png= 0 ⇒ x = y = 0.
  • Collinearity of points: Let ABC be three collinear points. Then, each pair of the vector 82650.png, is a pair of collinear vectors. Thus to check collinearity of three points, we can check the collinearity of any two vectors obtained with the help of three points.
  • Three points with position vectors, 82727.pngand 82723.png are collinear if and only if there exists three scalars xyz not all simultaneously zero such that 82718.png together with x + y + z = 0.

Non-collinear vectors

Two vectors acting in different directions are non-collinear vectors.
Non-collinear vectors are called independent vectors.

Free vectors

Vectors whose initial point is not specified are called free vectors.

Equal vectors

Two vectors are said to be equal, if they have the same magnitude and same direction.

Coplanar vectors

Vectors are said to be coplanar if all of them lie in the same plane. Three coplanar vectors are always dependent.
If vectors 79208.png, and 79216.png are coplanar then there exists scalars λ and μ such that 79222.png.
or 79263.png = 0
where ai, bi, ci (I = 1, 2, 3) are components of vectors 79237.png, and 79243.png in the direction of x, y, and z axis.
Four points with position vectors 79249.png are coplanar if 79256.png with λ1 + λ2 + λ3 + λ4 = 0.

Position vector of a point

If a point O is fixed in a space as origin then for any point P, the vector 79269.png is called the position vector (PV) of “P” w.r.t. “O”.
Also 79278.png = 79284.png = PV of BPV of A.

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