Simple Vector Component Projection

[oddjobmj]

Homework Statement

Consider the two vectors A=ai and B=3i+4j. What must be the value of a if the component of A along B is 6?

Homework Equations

The Attempt at a Solution

I've arrived at the correct answer by finding the angle between the x component of B (3) and B itself which comes out to 53.13 degrees. Then 6/cos53.13 = 10 which is the correct result. However, I still don't understand what it means for something to be the component of A along B. I was hoping someone here could shed some light on what this means and how to visualize it.

Thank you,
Odd

 

[Hootenanny]

Read this from Wikipedia and in particular the section on Scalar Project.

 

[BruceW]

Hootenanny, do you mean scalar product? If so, then yeah, I agree. The sentence 'component of A along B' is simply the scalar product of the vectors A and the unit vector of B, ie B/ lBl.

The scalar product can be written out in terms of the components of the two vectors as: (A₁B₁+A₂B₂+A₃B₃) / lBl (when the vectors are expressed in some orthogonal coordinate system).

For a geometrical representation, the scalar product of A and the unit vector B / lBl is also equal to lAl cos(angle) where angle is the angle between those vectors.

The intuitive way to think of 'the component of A along B' is that you simply look at the length of the vector A in the same direction as B.