Zero Vector, Unit Vector, Coinitial Vectors, Collinear Vectors, Equal Vectors, Negative of a Vector (Free Vector)
Zero Vector: A vector whose initial and terminal points coincide, is called a zero vector (or null vector), and denoted as `vec 0` . Zero vector can not be assigned a definite direction as it has zero magnitude. Or, alternatively otherwise, it may be regarded as having any direction. The vectors `vec ("AA") , vec (BB)` represent the zero vector,
Unit Vector: A vector whose magnitude is unity (i.e., 1 unit) is called a unit vector. The unit vector in the direction of a given vector `vec a` is denoted by `hat a`.
Coinitial Vectors: Two or more vectors having the same initial point are called coinitial vectors.
Collinear Vectors: Two or more vectors are said to be collinear if they are parallel to the same line, irrespective of their magnitudes and directions.
Equal Vectors: Two vectors are said to be equal, if they have the same magnitude and direction regardless of the positions of their initial points, and written as `vec a =vec b`.
Negative of a Vector: A vector whose magnitude is the same as that of a given vector , but direction is opposite to that of it, is called negative of the given vector.
For example, vector `vec (BA)` is negative of the vector `vec (AB)` , and written as `vec (BA) = - vec (AB)`.
Remark: The vectors defined above are such that any of them may be subject to its parallel displacement without changing its magnitude and direction. Such vectors are called free vectors.
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If `veca` and `vecb` are non- collinear vectors, find the value of x such that the vectors `barα = (x - 2)veca + vecb` and `barβ = (3+2x)bara - 2barb` are collinear.
Find the value of λ for which the four points with position vectors `6hat"i" - 7hat"j", 16hat"i" - 19hat"j" - 4hat"k" , lambdahat"j" - 6hat"k" "and" 2hat"i" - 5hat"j" + 10hat"k"` are coplanar.