Could somebody please help me with these Physics questions?

Question

Two vectors are directed at an angle α below the horizontal, as shown.

If the grid is 3.00 cm on a side, each vector has a magnitude 12.0 cm. Estimate the magnitude of C→ .

Two vectors are directed at an angle α below the horizontal, as shown.

 

If the grid is 5.00 cm on a side, each vector has a magnitude 20.0 cm. Estimate the magnitude of D→ .

Answer 1

You can estimate these graphically by drawing the addition as head to tail like this:

where you move the B vector so it starts at the end of A then you measure from the start of A to the end of B. You get about 4.1.

You can also do this by estimating the "y-value" of the vector (i.e., the amount the vector goes down across the 3 cm grid). It looks to me as if it measures about 2 cm out of the 3 cm grid. That means when I put vector B at the tip of A, vector B would drop another 2 cm to make the sum equal to 4 cm which matches pretty well with the other method.

For D = A - B, that is equivalent to D = A + -B and -B means vector B would go up and to the left instead of down and to the right and adding would look like this:

We would need to "remeasure the vertical distance (y-direction) using a 5 cm grid and I got about 4.1 cm

Using the Pythagorean Theorem, where "x" is the distance left on Vector A like this:

we calculate x like this:

4.1² + x² = 20²

x² = 383

x = 19.6

We would need to add another 19.6 to get the -B vector. 19.6 + 19.6 = 39.1 cm so D has a magnitude of about 39.1 cm or maybe round to 40 cm

Answer 2

A couple of ways to think about this:

The mathematical explanation is the following:

1. In both scenarios, A = -B and vice verse. B = -A.
2. So, since C = A + B. That also means C = A - A = 0
3. Similarly, D = A - B. D = A - (-A) = A + A = 40cm (this is positive because it asks for magnitude)

The logical conclusion is to think about it this way:

1. The vectors are the same magnitude, but opposite directions in both problems.
2. For C = A + B, that means they are going to cancel each other out. Just like if you were to pull on a rope with the same amount of force from both sides. So, C = 0.
3. For D = A - B, that means both vectors are going to be added in the same direction due to them originally being opposite. So, it is twice the original vectors' magnitudes. 2 x 20cm = 40cm