ree Vectors & Position Vectors

A vector that has no specific position in space is called a free vector.

Consider the position of a point A relative to a fixed origin O.  OA is called the position vector of A relative to O.  This displacement is unique and cannot be represented by another line.

Note:  Any vector AB can be written in terms of the position vectors of A and B:

AB = OB - OA

calar Multiplication

Let a be a non-zero vector and k be any real number. We define a new vector, ka, as follows.

(a) If k = 0, the ka = 0.

(b) If k > 0, the ka has the same direction as a and |ka| = k|a|.

(c) If k < 0, the ka is in the opposite direction to that of a and |ka| = -k|a|.

For all vectors a and b and all scalars l and m, we have:

(a) l(ma) = (lm)a,

(b) (l ± m)a = la ± ma,

(c) l(a ± b) = la ± lb.

nit Vectors

(a)  A unit vector is a vector with a magnitude of one unit.  A unit vector in the direction of a will be typed as a^.

(b)  Any vector r can be expressed as the product of its magnitude and the unit vector in the same direction: r = |r|r^
(c) Three important unit vectors are defined as follows:

atio Theorem

 

If a point P divides a line segment AB in the ratio m : n, then 
mb + na
OP = ¾¾¾¾
m + n
where a and b are the position vectors of A and B w.r.t. the origin respectively.
  

Note:  If M is the mid-point of AB, then OM = (OA + OB)/2.

3 Vectors

If P(a, b, c) is any point in 3-D Cartesian space, then the position vector of the point P is OP = p where
 
æ a ö
(a) p = ai + bj + ck or  ç b ÷
è c ø

(b) |p| = (a2 + b2 + c2)1/2 = d
 
æ a/d ö
(c) p^ = p/|p| =  ç b/d ÷
è c/d ø 

ome Results

A    Parallel

Let a and b be non-zero vectors. 
Then a is parallel to b if and only if
a = kb for some scalar k.

B    Collinear

Three points A, B and C are collinear if and only if
AB = kAC for some scalar k.

C    Coplanar

Let a and b be non-zero, non-parallel vectors.
A vector c is coplanar with a and b if and only if
c = la + mb for some scalars l and m.

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