7.5 Scalar and Vector Projections
Recall: For non-zero vectors a and b we define the dot product as:
to find angle between a and b:
Vectors are used in computer animation to determine the length of a shadow projected onto a
flat surface. We use this idea to define the projection of a vector on a vector.
Let a = OA and b = OBbe any two vectors forming an angle θ.
Let N be the point on the line OB such that AN ⊥ OB
We define the projection of a on b to be the vector ON
Magnitude of Projection:
Also called Scalar Projection
Scalar Projection
Scalar projection of a on b :
Scalar projection of b on a :
Vector Projection
Recall that a unit vector is a vector 1 unit in length. Therefore a unit vector in the direction of b :
Vector Projection of a on b
or
Ex1
u and v act at 60o to each other. If u =20N and , v =40N, determine:
a) Scalar projection
of u on v
b) Vector projection
of u on v
c) Vector projection
of v on u
Ex2 If u and v act a 150o to each other, and if u = 20 and v = 40, determine:
a) Vector projection of u on v
b) Sketch the projection
Ex3 If u =[1,5] and v=[-4,2], find the scalar and vector projections of v onto u.
.
Direction Cosines for OP = [a, b, c]
If
are the angles that OP makes with the
positive x-axis, y-axis and z-axis, respectively, then
Ex.4: If determine:
a)
b) Direction cosines of
c) Direction angles of 7.5 Assignment: P.398 #398 # 1 6 11ab 12 13a 15b