Any numbers proportional to the direction cosines of a line are called the di-
rection numbers of the line. Show for a:b:cthe direction numbers of a line which
are not all zero, then the direction cosines are given by
cos α=a
rcos β=b
rcos γ=c
r,where r=
√
a^2 +b^2 +c^2.Example 6-2.
Sketch a large version of the letter H. Con-
sider the sides of the letter H as parallel lines a
distance of punits apart. Place a unit vector ˆe
perpendicular to the left side of H and pointing
toward the right side of H. Construct a vector x 1
which runs from the origin of eˆto a point on the
right side of the H. Observe that ˆe·x 1 =pis a
projection of x 1 on ˆe.Now construct another vector x 2 ,different from x 1 ,again from
the origin of eˆto the right side of the H. Note also that eˆ·x 2 =pis a projection
of x 2 on the vector eˆ.Draw still another vector x, from the origin of ˆeto the right
side of H which is different from x 1 and x 2 .Observe that the dot product ˆe·x =p
representing the projection of xon ˆestill produces the value p.
Assume you are given ˆeand pand are asked to solve the vector equation ˆe·x =p
for the unknown quantity x. You might think that there is some operation like vector
division, for example x =p/eˆ,whereby x can be determined. However, if you look
at the equation ˆe·x =pas a projection, one can observe that there would be an
infinite number of solutions to this equation and for this reason there is no division
of vector quantities.
ComponentFormforDotProduct
Let A, B be two nonzero vectors represented in the component form
A=A 1 eˆ 1 +A 2 eˆ 2 +A 3 ˆe 3 , B=B 1 ˆe 1 +B 1 ˆe 2 +B 3 ˆe 3The dot product of these two vectors is
A·B= (A 1 ˆe 1 +A 2 ˆe 2 +A 3 ˆe 3 )·(B 1 eˆ 1 +B 1 ˆe 2 +B 3 ˆe 3 ) (6 .14)