How Do You Calculate Distance and Displacement in Vector Calculus?

[spoc21]

Homework Statement

The displacements of two ships, A and B, two hours after leaving from the same port can be represented with position vectors $$\vec{OA}$$ [20, 50, 0] and
($$\vec{OB}$$) [60, 10, 0]. Assume that the port is located at the origin and that all units are in kilometres.

a. How far from the port is each ship?

b. How far apart are the two ships?

These subparts are part of the same question:

The displacement of a bird from the port can be described with the vector
-65 i – 8 j + 0.5 k

i) How high above the water is the bird?


ii) How far from ship B is the bird?


d. What will be the position vector of the displacement of ship A from the port 3.5 hours after leaving the port? Assume that the direction and speed of the ship are constant.

Homework Equations

The Attempt at a Solution

For qu a, I just took the magnitude of the vectors OA, and OB, using the magnitude formula, getting 53.85 km, and 60.83 km for the distances from the port. I was just wondering if this method is correct?

For b, I'm confused, would I just add the two vectors OA and OB, and then find the magnitude of the resultant vector , I'm 90% that this method is correct, but would appreciate any helpful tips.

For qu c, part 1, I'm thinking that the first part is the same as a), since we just calculate the magnitude of displacement vector from the origin (but am very unsure about this)

for qu c), part 2, I am very confused, and tips to help me get started would be greatly appreciated.

For qu d), I am thinking that we just divide the displacement vector of ship B by 1.75, in order to get the position vector for the ship after 3.5 hours. Again, I am very unsure about this.

I would really appreciate any help,

thanks.

 

[CompuChip]

If you want to know the vector that points from one object to another (let's call that a displacement or difference vector), you need to subtract them. So if O[0, 0 0] is the origin, then
OA = A - O = [20, 50, 0] - [0, 0, 0] = [20, 50, 0]
is the vector that points from the origin to A, and for example
AB = B - A
is the vector that points from A to B.

For c.i) it may help to make a drawing. Sketch the axes, origin, position of the bird, and indicate in the picture which distance is being asked.

c.ii) is solved in the same way as b) (you can again check this in the picture, they are asking for the magnitude of a displacement vector).

For d), I think your answer is correct but I can't see how you arrived at it. If you want to do it systematically, you can find the vectorial displacement of the ship in one hour (since its units will be distance per time, i.e. km/h, this is actually the velocity vector). Then what is the position 5.5 hours after leaving port?

 

[spoc21]

Thanks compuchip..Im still confused about ci), finding the height of the bird..how would it be possible to find the height of the bird?? thanks

 

[spoc21]

bump..anyone?

 

[CompuChip]

If you draw the picture, you will see that the height above the ground is simply the z-coordinate of the vector.