Multiplying a Vector by a Scalar

5. Multiplying a Vector by a Scalar:

If k is any positive scalar and a is a vector then ka is a vector in the same direction as a but k times longer. If k is negative, ka is a vector in the opposite direction to a and k times longer. The vector ka is said to be a scalar multiple of a. The vector 3a is three times as long as a and has the same direction. The vector 1 2r is in the same direction as r but is half as long. The vector −4b is in the opposite direction to b and four times as long. For any scalars k and l, and any vectors a and b the following rules hold:
if k is positive if k is negative.

Key Point:
k(a + b)=ka+ kb
(k + l)a = ka+ la
k(l)a =(kl)a

Exercise For You:

Now do this exercise Using the rules given in the previous Key Point, simplify the following: a) 3a +7a b) 2(7b) c) 4q +4 r

Unit vectors:

A vector which has a magnitude of 1 is called a unit vector. If a has magnitude 2, then a unit vector in the direction of a is 2a.

A unit vector has length one unit. A unit vector in the direction of a given vector is found by dividing the given vector by its magnitude.
A unit vector in the direction of a is given the ‘hat’ symbol ˆ a.
Key Point A unit vector can be easily found by dividing a vector by its modulus. ˆ
a =a |a|

More exercises for you to try:

 1. Draw an arbitrary vector r. On your diagram draw 2r,4 r, −r, −3r and 1 2r. 2. In triangle OAB the point P divides AB in the ratio m : n. If−→ OA = a and −→ OB = b depict this on a diagram and then find an expression for −→OP in terms of a and b .