### 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:

i is a unit vector in the +ve direction of x-axis,
j is a unit vector in the +ve direction of y-axis,
k is a unit vector in the +ve direction of z-axis.

### atio Theorem

If a point P divides a line segment AB in the ratio m : n, then�
 mb + na OP = ���� 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
 � a � (a) p = ai + bj + ck or� � b � � c �

(b) |p| = (a2 + b2 + c2)1/2 = d
 � a/d � (c) p^ = p/|p| =� � b/d � � c/d ��

### 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 (or more) points are said to be collinear if they lie on the same straight line.

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

C��� Coplanar

Three (or more) vectors are said to be coplanar if they lie on the same plane.

 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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