Finding the hypotenuse

Question

I have the question "The force vectors in the following diagrams are all coplanar but not drawn to scale. Use appropriate trigonometry to answer the following questions.

Calculate the resultant force on the following objects and the acceleration it produces."

For this I have made a triangle and have used Pythagoras to find the length of the hypotenuse R. The answer is get for this is:

However, the solutions say that the answer for the length R should be 4.2 N.

I do not understand how This is achieved.

Answer 1

First combine two horizontal forces. As they are in opposite direction. So you got 3 N as resultant force. Now,

$\sqrt{(3)^2 + (3)^2} = \sqrt{18}$ = 4.24 N

Answer 2

Two things:

First (and most importantly in this case), you ignored the $4$ Newton vector. You're supposed to find the resultant force of all three of the forces shown. Fortunately, the $4$ and $7$ are along the same line (although in opposite directions) so you don't need trigonometry or Pythagoras to figure out how to combine them.

Second, the way to find the resultant of two vectors is not generally by drawing an arrow from the tip of one vector to another. You can put the two vectors tip-to-tail and then go from the first tail to the second tip, or make a parallelogram and draw the diagonal from the common tail of the vectors. When the angle between the vectors is a right angle you get the correct magnitude anyway (since both diagonals of a rectangle are equal) but you do not get the correct direction. (It's unclear from the question whether you needed to find the direction of acceleration, but at least sometimes you will need to know how to do it.)

The easy way to do this problem is to combine the two horizontal vectors first, which gives you a (smaller) horizontal vector: the force of $4$ N to the right partially cancels the $7$ N to the left; $7 - 4 = 3,$ so the net force is $3$ N to the left. Then combine that with the vertical vector. That way you will have accounted for all three vectors.