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Unit Vectors
Base Vectors and Vector Components
Negative of a Vector
Some Key Points
Some Basic Rules of Vectors
Vectors constitute an important topic in the Mathematics syllabus of JEE. It is important to master this topic to remain competitive in the IIT JEE. It often fetches some direct questions too. The topic of Vectors is quite simple and it also forms the basis of several other topics.
Various topics that have been covered in this chapter include:
Introduction
Addition of Two Vectors
Fundamental Theorem of Vectors
Orthogonal System of Vectors
We now discuss some of these topics in brief as they have been covered in detail in the coming sections:
Length / Magnitude of a vector: The length or magnitude of the vector w = (a, b, c) is defined as
|w|= w= √a^{2}+b^{2}+c^{2}
A unit vector is a vector of unit length. A unit vector is sometimes denoted by replacing the arrow on a vector with a "^" or just adding a "^" on a boldfaced character (i.e., ? or ?). Therefore,
| ?| = 1.
Any vector can be made into a unit vector by dividing it by its length.
? = u / |u|
Any vector can be fully represented by providing its magnitude and a unit vector along its direction. A vector can be written as u = u?
You may refer the Sample Papers to get an idea about the types of questions asked.
Base vectors represent those vectors which are selected as a base to represent all other vectors. For example the vector in the figure can be written as the sum of the three vectors u_{1}, u_{2}, and u_{3}, each along the direction of one of the base vectors e_{1}, e_{2}, and e_{3}, so that
u= u_{1}+u_{2}+u_{3}
It is clear from the figure that each of the vectors u_{1}, u_{2} and u_{3} is parallel to one of the base vectors and can be written as a scalar multiple of that base. Let u_{1}, u_{2}, and u_{3} denote these scalar multipliers such that one has
u_{1}= u_{1}e_{1}
u_{2}=u_{2}e_{2}
u_{3}=u_{3}e_{3}
The original vector u can now be written as
u= u_{1}e_{1}+u_{2}e_{2}+u_{3}e_{3}
Watch this Video for more reference
A negative vector is a vector that has the opposite direction to the reference positive direction.
A vector connecting two points:
The vector connecting point A to point B is given by
r= (x_{B}-x_{A}) i+ (y_{B} – y_{A}) j + (z_{B} – z_{A})k , here i, j and k denote the unit vectors along x, y and z axis respectively.
The magnitude of a vector is a scalar and scalars are denoted by normal letters.
Vertical bars surrounding a boldface letter denote the magnitude of a vector. Since the magnitude is a scalar, it can also be denoted by a normal letter; |w| = w denotes the magnitude of a vector
The vectors are denoted by either drawing a arrow above the letters or by boldfaced letters.
Vectors can be multiplied by a scalar. The result is another vector.
Suppose c is a scalar and v = (a, b) is a vector, then the scalar multiplication is defined by cv= c (a,b)= (ca,cb). Hence each component of a vector is multiplied by the scalar.
Then the sum of these two vectors is defined by
v + u = (a + e, b + f, c + g).
v - u = v + (-1)u.
If u, v and w are three vectors and c, d are scalars then the following hold true:
u + v = v + u (the commutative law of addition)
u + 0 = u
u + (-u) = 0 (existence of additive inverses)
c (du) = (cd)u
(c + d)u = cu +d u
c(u + v) = cu + cv
1u = u
u + (v + w) = (u + v) + w (the associative law of addition)
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