Calculus 3 vector question

[bfaweio]

1. A ship at position (1,0) on a nautical chart (with north in the positive y direction sights a rock at position (2,4). What is the vector joining the ship to the rock? What angle does this vector make with due north?

I said the vector was at (1,4) and by drawing the diagram, I found the horizontal distance 1 and the vertical distance 4. I found the angle to be tan^(-1) of 4, or 1.33 rad. Is this right?

2. Suppose that the same ship is pointing due north and traveling at a speed of 4 knots relative to the water. There is a current flowing due east at 1 knot. (1 knot= 1 nautical mile/hr)

a. If there were no current, what vector would represent the velocity of the ship relative to the ship bottom?

I drew the vector pointing north and since it is going 4 knots north, I think the vector would be at (0,4).

b. If the ship were just drifting with the current, what vector v would represent its velocity to the sea bottom?

c. What vector represents the total velocity of the ship?

I don't understand the difference between the last two questions. I thought b would be (0+1, 4), or (1,4) but c looks like it would be the same answer as well.

Thanks

 

[bfaweio]

Just kidding I figured it out

 

[HallsofIvy]

For the benefit of others who may have wondered

1. A ship at position (1,0) on a nautical chart (with north in the positive y direction sights a rock at position (2,4). What is the vector joining the ship to the rock? What angle does this vector make with due north?

I said the vector was at (1,4) and by drawing the diagram, I found the horizontal distance 1 and the vertical distance 4. I found the angle to be tan^(-1) of 4, or 1.33 rad. Is this right?

The vector is (1, 4). Drawing that, a line due north and drawing a perpendicular from the tip of the vector to the line north, we have right triangle with "opposite side" of length 1 and near side of length 4 so the angle is arctan(1/4), not arctan(4). arctan(4) is the angle the vector makes with due East.

2. Suppose that the same ship is pointing due north and traveling at a speed of 4 knots relative to the water. There is a current flowing due east at 1 knot. (1 knot= 1 nautical mile/hr)

a. If there were no current, what vector would represent the velocity of the ship relative to the ship bottom?

I drew the vector pointing north and since it is going 4 knots north, I think the vector would be at (0,4).

Yes, that is correct.

b. If the ship were just drifting with the current, what vector v would represent its velocity to the sea bottom?

Much the same as (a) except now we use the current's velocity and not the boats: (1, 0).

c. What vector represents the total velocity of the ship?

Now, we take the sum of the two velocities: (1, 0)+ (0, 4)= (1, 4).

I don't understand the difference between the last two questions. I thought b would be (0+1, 4), or (1,4) but c looks like it would be the same answer as well.

Thanks

"drifting with the current" means NOT using its own power. That's why I dropped the boat's velocity.