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u = [1537]and v = [2468]

Solution: Type and execute each of the following commands, while interpret-
ing each output:

u=[1 5 3 7]
v=[2 4 6 8]
u*v'
v'*u
u*v %you cannot multiply two rows
u'*v
u*u'
(norm(u))^2
As observed from the above results, in MATLAB, the dot product can be
obtained only by the multiplication of a row on the left and a column of the
same length on the right. If the order of a row and column are exchanged, we
obtain a two-dimensional array structure (i.e., a matrix, the subject of Chap-
ter 8). On the other hand, if we multiply two rows, MATLAB gives an error
message about the non-matching of dimensions.
Observe further, as pointed out previously, the relation between the length
of a vector and its dot product with itself.

In-Class Exercises

Pb. 7.2 Generalize the analytical technique, as previously used in Example
7.4 for finding the normal to a line in 2-D, to find the unit vector in 3-D that
is perpendicular to the plane:

ax + by + cz + d = 0

(Hint: A vector is perpendicular to a plane if it is perpendicular to two non-
collinear vectors in that plane.)
Pb. 7.3 Find, in 2-D, the distance of the point P(x , y ) from the line ax + by
0
0
+ c = 0. (Hint: Remember the geometric definition of the dot product.)
Pb. 7.4 Prove the following identities:

r r r r r r r r r r r r r r r
uv⋅= v u⋅ , u v⋅( + w =) u v u w⋅ + ⋅ , k uv⋅( ⋅ ) = ( ku v⋅)

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