I have a Force Vector $= 100N$, making an Angle $= 45$ degrees. When I find their Components trigonometrically, I get $70N$ each; as
$$F_x = 100\cos(45) = 70N$$ $$F_y = 100\sin(45) = 70N$$
Verifying the result, by Head-to-Tail method, I get $70N + 70N = 140N$.
Why is there descripancy or where am I making a mistake? Please help.
On
You have a component of force in the x direction and a component in the y direction when you add them together the components each maintain their direction.
you could write this as $70\mathbf i + 70 \mathbf j$ or $(70,0) + (0,70) = (70,70)$ depending on which notation you choose to use.
You can't just add the magnitudes of vectors in different directions and get the magnitude of the sum.
In fact we have the "triangle inequality" which says. $\|F_x + F_y\|\le \|F_x\|+\|F_y\|$
The magnitude of the vector is $\|F_x + F_y \| = \sqrt {F_x^2 + F_y^2}$
On
When you add vectors using the head-to-tail method, you still have to maintain the correct directions of the vectors. When you add the two component vectors that you’ve computed using this method, you get something like this:
Since the two dotted vectors are perpendicular, the length of the resulting vector is given by the Pythagorean theorem: $\sqrt{70^2+70^2}\approx 99$. This isn’t exactly $100$ because you truncated the actual value of the length of the components ($50\sqrt2 = 70.71\dots$) to $70$. If you use the better approximation of $70.7$, the resulting length is much close to the original $100$.
What you effectively did was to add two vectors that point in the same direction, which is not what you’ve got after decomposing the original force vector.
Pretending that $\frac 12\sqrt 2=0.7$, which is not correct, your force vector is $F=(70,70)$. You have resolved it correctly into $F_x=(70,0)$ and $F_y=(0,70)$. When you add these together you add the components and get back to $(70,70)$ as you should. The fact that the sum of the magnitudes is $140$ is not important.