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Example: Try this. Stand facing a partially open door with the front of your body parallel to the plane of the door (a door with no springs is best). Hold the outer edge of the door with one hand. Press down and note that the door is not opened or closed. Push towards the hinge and note that the door is not opened or closed. Push and pull away and towards your body and note how easily you cause the door to rotate. Thus the only force component that tends to rotate the door is perpendicular to the plane of the door (which is the plane of the hinge and line from the hinge to your hand).
We have Then we find ˆ·λ thus turned our vector equation into a scalar equation and eliminated the unknown N at the same time. 19: The dot product helps find components in terms of crooked unit vectors. For example, A y = A·ˆ = Ax (ıˆ ·ˆ ) + Ay (ıˆ ·ˆ ) = Ax (− sin θ ) + Ay (cos θ ). a) 26 CHAPTER 2. Vectors for mechanics Using dot products to solve geometry problems We have seen how a vector can be broken down into a sum of components each parallel to one of the orthogonal base vectors. 20: For any A and λˆ , A can be ˆ and a decomposed into a part parallel to λ ˆ part perpendicular to λ.
If A is multiplied by 7 than so must be each of the component vectors. Thus [cA]x yz = [c A x , c Ay , c A z ]. The components of a scaled vector are the corresponding scaled components. Magnitude of a vector using components The Pythagorean theorem for right triangles (‘A2 + B 2 = C 2 ’) tells us that |F | = Fx2 + Fy2 , (2D) |F | = Fx2 + Fy2 + Fz2 . (3D) To get the result in 3D the 2D Pythagorean theorem needs to be applied twice successively, first to get the magnitude of the sum F x + F y and once more to add in F z.