I don't understand the solution to this question:
Find the component of the force in the direction of the vector .
Let . Then the unit vector in the direction of is .
Hence the component of in the direction of is
What do the expressions and mean? How in general do you solve these problems? Thanks
I'm going to try to explain the geometric meaning of a dot product.
If you have 2 vectors , which include the angle and you know to calculate the length of a vector then the dot product is defined as:
That means is the "shadow" of vector b on the vector a (if you use a light with parallel rays perpendicular to vector a)
The value of the dot product is a real number. This number represents the area of a rectangle (painted red in my sketch)
If you use a unit vector of a ( ) then the value of the area (in square units) is as great as the "shadow" of b (in length units) with respect to the vector a. (compare the rectangle painted blue)
That means: With the use of an unit vector you have introduced a kind of metric and you can calculate the length of components with respect to given directions.
And now it must be quite clear why is perpendicular to if