vector with diagram

Question

A lighthouse is at the position of 7i - 3j , relative to the wharf. A boat is travelling in the direction parallel to 3i - j from the wharf. IF all measurements are in km, find to the nearest meter, the closest distance the boat will get to the lighthouse if it does not change directions.

State any assumption and effect of the assumption.


Can anyone please help me find the answer to this vector question with a diagram to assist?

Answer 1

Let's use regular coordinate geometry to make a diagram and figure this out.

 

The wharf is at the origin, the light house is at (7,-3) and the boat is traveling from the wharf in the direction 3 to the right and one down which is equivalent to a slope of -1/3. 

 

So the boat is traveling along the equation y= -1/3 x. So we have to find the point on this line that is closest to (7,-3). The minimum distance from a point to a line is the perpendicular drawn from the point to the line. So perpendicular to your line must have a slope of 3.....so the equation of the line using the point slope formula is 

 

y + 3 = 3(x - 7) which is y = 3x -24 when simplified.

 

We want to find where these 2 lines intersect, so,

-1/3 x = 3x -24 

x=7.2

making the coordinate (7.2,-2.4) the closest point that the ship will be to the lighthouse. 

 

Now find the distance between (7.2,-2.4) and (7,-3) and you'll have your answer.....sqrt(.40) =.632455532 km. To the nearest meter this is 632 meters.

Answer 2

by setting up a proportion 3/1 =7/x, we find the boat will be at position 7i - 7/3j which is north of the lighthouse. Therefore it can get closer and even hit the lighthouse if it follows a path parallel to 3i - j. The equation of the line for 3i - j has a slope = -1/3 and y intercept = 0

If you translate the line downward about 662 meters, and the boat follows this parallel line, it will hit the lighthouse. Or maybe I'm misinterpreting the question.

Answer 3

by setting up a proportion 3/1 =7/x, we find the boat will be at position 7i - 7/3j which is north of the lighthouse. Therefore it can get closer and even hit the lighthouse if it follows a path parallel to 3i - j. The equation of the line for 3i - j has a slope = -1/3 and y intercept = 0

If you translate the line downward about 667 meters, and the boat follows this parallel line, it will hit the lighthouse. Or maybe I'm misinterpreting the question.