Solving a Vector Problem in the xy Plane

[jpodo]

Here is the problem I can't solve:

You are given a vector in the xy plane that has a magnitude of 80.0 units and a y component of -60.0 units.

(a) What are the two possibilities for its x component?

(b) Assuming the x component is known to be positive, specify the vector which, if you add it to the original one would give a resultant vector that is 70.0 units long and points entirely in the -x direction. (looking for Magnitude + Direction)

 

[genneth]

Suppose that a vector has the components x and y, and a magnitude r. What's the equation that links x, y and r?

 

[m2003]

I would approach this problem by first visualizing the given vector in the xy plane. By knowing the magnitude and y component, I can draw a right triangle where the hypotenuse is 80.0 units and the opposite side is -60.0 units. From this, I can use the Pythagorean theorem to solve for the missing side, which would be the x component. This leads to two possibilities for the x component: +64.3 units and -64.3 units.

Next, to find the vector that, when added to the original one, would give a resultant vector that is 70.0 units long and points entirely in the -x direction, I would use basic vector addition principles. Since the resultant vector must point in the -x direction, the x component of the added vector must be negative. By setting up an equation using the magnitude of the resultant vector and the known x component of the added vector, I can solve for the y component. This would result in a vector with a magnitude of 70.0 units and a y component of -56.2 units, pointing in the -x direction.

In summary, to solve this vector problem in the xy plane, I would use visualization and basic vector addition principles to find the two possibilities for the x component and the vector that would give the desired resultant vector. This approach follows the scientific method of using observation, experimentation, and logic to solve problems.