✍️Revision Notes on Vectors
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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)
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) = a2 + 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 + μb1 and r = a2 + μb2 be two lines.
They intersect if (b1 x b2)(a2 - 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 + a2b2 + ..., + anbn .
