MULTIPLICATION of a VECTOR by a SCALAR
(Geometric Vectors)
A.
MULTIPLYING VECTORS by a SCALAR
Consider the vector, 𝑘𝑎 , where k is a scalar and 𝑎 is a nonzero vector:
1.
If k > 0, then 𝑘𝑎 is in the same direction as 𝑎 with magnitude 𝑘 𝑎 .
2.
If k < 0, then 𝑘𝑎 is in the opposite direction as 𝑎 with magnitude 𝑘 𝑎 .
NOTE: If k = 0, then 𝑘𝑎 = 0 (the zero vector).
B.
COLLINEAR VECTORS
Two vectors, 𝑢 and 𝑣, are collinear if and only if it is possible to determine a
nonzero scalar, k, such that 𝑢 = 𝑘𝑣 , k  0.
Ex.
Collinear vectors are
parallel or lie on the
same straight line.
Ex.
Express vector 𝑎 in terms of vector 𝑏:
𝑎
𝑏
C.
UNIT VECTOR
A unit vector, 𝑢, is a vector of magnitude 1 in the same direction as 𝑢.
𝑢=
1
𝑢
𝑢
Ex.
𝑢 =3
1
𝑢
𝑢= 𝑢
3
𝑢
D.
EXAMPLES
1.
An airplane is flying in the direction N30oE at an airspeed of 240 km/h. The
velocity for this airplane is represented by 𝑣. State the magnitude and
1
direction of the vector − 𝑣.
3
N
W
E
S
2.
If 𝑥 and 𝑦 are unit vectors that make an angle of 120o with each other:
a)
calculate 𝑥 − 2𝑦 ;
b)
determine a unit vector in the same direction as 𝑥 − 2𝑦.
Homework: p.298–301 #1–5, 8–10, 13–15, 17(T), 18(T)