Vectors and Dot Products

Let u = < u1 , u2 > and v = < v1 , v2 >.
The dot product of u and v is u × v = u1v1 + u2v2.
Notice, the dot product is a scalar (i.e., a number as opposed to a vector quantity).
 

Properties of the Dot Product

u and v are vectors; c is a scalar.
1.  u × v = v × u   (commutative)
2.  0 × v = v
3.  u × (v + w) = (u × v) + (u × w)  (dot product is distributive over addition)
4.  v × v = ||v||2
5.  c (u × v)=cu × v = u × cv (scalar multiplication is distributive over dot product)
To find the angle between two vectors, use cos q   =     u× v
                                                                                  ||u|| ||v||

Also, u × v = ||u|| ||v|| cos q   is an identity taken from the above formula.

Orthogonal vectors are vectors that are perpendicular to each other.
Let u and v be orthogonal vectors.
Then  u × v = 0
The zero vector is orthogonal to every vector because 0 × v = 0.

Let u and v be nonzero vectors such that u = w1 + w2 where w1 and w2 are orthogonal and w1 is parallel to v.
w1 and w2 are vector components of u.
w1 is the projection of u onto v and is denoted by w1 = projvu.
The vector w2 is given by w2 = u – w1.

The projection of u onto v is given by projvu = (    u× v  ) v
                                                                             ||v||2
                                                                                                                                     __
The work done by a constant force F as its point of application moves along the vector PQ is given by either of the following:

1.  W = ||projPQF|| ||PQ||        (projection form)

2.  W = F × PQ   (dot product form)  (This is the same as Fd)
 

Problems

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