Revision Notes - Vectors

• The length or the magnitude of the vector  = (a, b, c) is defined by w = √a2+b2+c2
• A vector may be divided by its own length to convert it into a unit vector, i.e. ? = u / |u|. (The vectors have been denoted by bold letters.)
• If the coordinates of point A are xA, yA, zA and those of point B are xB, yB, zB then the vector connecting point A to point B is given by the vector r, where r = (xB - xA)i + (yB – yA) j + (zB – zA)k , here i, j and k denote the unit vectors along x, y and z axis respectively.
• Some key points of vectors:
1) The magnitude of a vector is a scalar quantity
2) Vectors can be multiplied by a scalar. The result is another vector.
3) 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 vector is multiplied by the scalar.
4) If two vectors are of the same dimension then they can be added or subtracted from each other. The result is gain a vector.
• If u, v and w are three vectors and c, d are scalars then the following results of vector addition hold true:
1) u + v = v + u (the commutative law of addition)
2) u + 0 = u
3) u + (-u) = 0 (existence of additive inverses)
4) c (du) = (cd)u
5) (c + d)u = cu + d u
6) c(u + v) = cu + cv
7) 1u = u
8) u + (v + w) = (u + v) + w (the associative law of addition)
• Given two vectors u and v, their sum or resultant written as (u+v) is also a vector obtained by first bringing the initial point of v to the terminal point of u and then joining the initial point of u to the terminal point of v giving a consistent direction by completing the triangle OAP. This is termed as the Triangle Law of Addition.
• Let a and b be any two vectors. From the initial point of a, vector b is drawn and the parallelogram AOCB is completed with OA and OB as adjacent sides. The vector OC is defined as the sum of a and b. This is called the parallelogram law of vectors.
• For adding more than two vectors, Polygon Law of Addition is used.
• Internal and External Division:
1) If A and B are two points with position vectors a and b respectively and C is a point which divides AB internally in the ratio m : n, then the position vector of C is given by OC = (mb + na)/ (m + n). This is termed as internal division.
2) If A and B are two points with position vectors a and b respectively and C is a point which divides AB externally in the ratio m : n, then the position vector of C is given by OC = (mb - na)/ (m-n)

3) In case, C is the mid-point of AB, then the position vector of C is (a + b)/2.
4) The position vector of any point C on AB can always be assumed as c = la + mb, where l + m = 1.
• If circumcentre is origin and vertices of a triangle have position vectors a, b and c, then the position vector of orthocentre will be -(a+b+c).
• a.b is the product of length of one vector and length of the projection of the other vector in the direction of former.
• a.b = |a| Projection of |b| in direction of a = |b|. Projection of |a| in direction of b.
• Some Basic Rules of Algebra of Vectors:
1) a.a = |a|2 = a2
2) a.b = b.a
3) a.0 = 0
4) a.b = (a cos q)b = (projection of a on b)b = (projection of b on a) a
5) a.(b + c) = a.b + a.c (This is also termed as the distributive law)
6) (la).(mb) = lm (a.b)
7) (a ± b)2 = (a ± b) . (a ± b) = a+ b2 ± 2a.b
8) If a and b are non-zero, then the angle between them is given by cos θ = a.b/|a||b|
9) a x a = 0
10) a x b = - (b x a)
11) a x (b + c) = a x b + a x c
• Any vector perpendicular to the plane of a and b is l(a x b) where l is a real number.
• Unit vector perpendicular to a and b is ± (a x b)/ |a x b|
• The position of dot and cross can be interchanged without altering the product. Hence it is also represented by [a b c]
1) [a b c] = [b c a] = [c a b]
2) [a b c] = - [b a c]
3) [ka b c] = k[a b c]
4) [a+b c d] = [a c d] + [b c d]
5) a x (b x c) = (a x b) x c, if some or all of a, b and c are zero vectors or a and c are collinear.
• Methods to prove collinearity of vectors:
1) Two vectors a and b are said to be collinear if there exists k ? R such that a = kb.
2) If p x q = 0, then p and q are collinear.
3) Three points A(a), B(b) and C(c) are collinear if there exists k ? R such that AB = kBC i.e. b-a = k (c-b).
4) If (b-a) x (c-b) = 0, then A, B and C are collinear.
5) A(a), B(b) and C(c) are collinear if there exists scalars l, m and n (not all zero) such that la + mb+ nc = 0, where l + m + n = o
• Three vectors p, q and r are coplanar if there exists l, m ? R such that r = lp + mq i.e., one can be expressed as a linear combination of the other two.
• If [p q r] = 0, then p, q and r are coplanar.
• Four points A(a), B(b), C(c) and D(d) lie in the same plane if there exist l, m ? R such that b-a = l(c-b) + m(d-c).
• If [b-a c-b d-c] = 0 then A, B, C, D are coplanar.
• Two lines in space can be parallel, intersecting or neither (called skew lines). Let r = a1 + μb and r = a2 + μb2 be two lines.
• They intersect if (b1 x b2)(a- a1) = 0
• The two lines are parallel if b1 and b2 are collinear.
• The angle between two planes is the angle between their normal unit vectors i.e. cos q = n1 . n2
• If a, b and c are three coplanar vectors, then the system of vectors a', b' and c' is said to be the reciprocal system of vectors if aa' = bb' = cc' = 1  where a' = (b xc) /[a b c] , b' = (c xa)/ [a b c] and c' = (a x b)/[a b c] Also, [a' b' c'] = 1/ [a b c]
• Dot Product of two vectors a and b defined by a = [a1, a2, ..., an] and b = [b1, b2, ..., bn] is given by a1b1  + a2b+ ...,  + anbn
• Some Basic Results of Vector Calculus:
1) Vectors in the same direction can be added by simply adding their magnitudes. But if the vectors to be added are in opposite directions, then their magnitudes are subtracted and not added.
2) Column vectors can be added by simply adding the values in each row.
3) You can find the magnitude of a vector in three dimensions by using the formula a2 = b+ c2 + d2, where a is the magnitude of the vector, and b, c, and d are the components in each direction.
4) If l1a + m1b = l2a + m2b then l1 = l2 and m= m2
5) Collinear Vectors are also parallel vectors except that they lie on the same line.
6) When two vectors are parallel, the dot product of the vectors is 1 and their cross product is zero.
7)Two collinear vectors are always linearly dependent.
8) Two non-collinear non-zero vectors are always linearly independent
9) Three coplanar vectors are always linearly dependent.
10) Three non-coplanar non-zero vectors are always linearly independent.
11) More than 3 vectors are always linearly dependent.
12) Three vectors are linearly dependent if they are coplanar that means any one of them can be represented as a linear combination of other two.
The document Vectors Revision Notes | JEE Main & Advanced Mock Test Series is a part of the JEE Course JEE Main & Advanced Mock Test Series.
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## FAQs on Vectors Revision Notes - JEE Main & Advanced Mock Test Series

 1. What is a vector in the context of JEE?
Ans. In the context of JEE, a vector refers to a quantity that has both magnitude and direction. It is represented by an arrow, where the length of the arrow represents the magnitude of the vector and the direction of the arrow represents the direction of the vector.
 2. What is the importance of vectors in JEE?
Ans. Vectors play a crucial role in JEE as they are used to represent various physical quantities such as displacement, velocity, acceleration, force, and more. Understanding vectors helps students solve problems involving these quantities and is essential for success in JEE.
 3. How are vectors represented in JEE?
Ans. Vectors in JEE are typically represented using their components along different axes. The components are usually represented as a tuple (x, y, z) for three-dimensional vectors or (x, y) for two-dimensional vectors. These components help in performing vector operations such as addition, subtraction, and scalar multiplication.
 4. Can vectors be added or subtracted in JEE?
Ans. Yes, vectors can be added or subtracted in JEE. Vector addition involves adding the corresponding components of two vectors, while vector subtraction involves subtracting the corresponding components. These operations are important for solving problems that require combining or comparing vectors.
 5. Are there any specific techniques or formulas to solve vector-related problems in JEE?
Ans. Yes, there are specific techniques and formulas to solve vector-related problems in JEE. Some commonly used techniques include using the parallelogram law of vector addition, dot product, cross product, and resolving vectors into components. Familiarity with these techniques and formulas can help students solve vector problems efficiently in JEE.

## JEE Main & Advanced Mock Test Series

366 docs|219 tests

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