How Does Vector Addition Apply in Calculating River Crossing Distances?

[dance_sg]

What is vector addition?

 

[RoyalCat]

Ooh, that's a big one.

Here are a few pages you might find useful:
http://en.wikipedia.org/wiki/Euclidean_vector
http://mathforum.org/library/drmath/sets/select/dm_vectors.html

A vector is a sort of mathematical entity. It has an amplitude (How big it is) and a direction (Where it's pointed). In some cases, there is also importance to where its origin is.

Two-dimensional vectors, are vectors that have a size and a direction in a two-dimensional plane. As such, each vector can be represented by two components, its x and y projections.

For instance, going 100 paces north-east, would get you to the same spot as would, going 70 paces east, and 70 paces north (Make a 1:10 scale drawing, and see for yourself this is true!).
So you could say, that the projections of the vector, (100 paces in the north-east direction) relative to the north-south, east-west plane, are (70 paces in the east direction) and (70 paces in the north direction).

I'll scrounge up a couple more links for you in a sec. Showing what vector sums are without being able to draw is pretty hard.
EDIT:
Here you go: :)
http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=51
http://hyperphysics.phy-astr.gsu.edu/hbase/vect.html
http://mathworld.wolfram.com/VectorAddition.html

 

[DavidSnider]

Vectors have length and direction and they describe a relative displacement from something. So (1,2) would be 1 unit across and 2 units up

You add 2 Dimensional vectors like this:
(a,b) + (c,d) = (a + c, b + d)

and it's the same for N-Dimensional vectors
(a,b,c) + (d,e,f) = (a + d, b + e, c + f) , etc..

 

[dance_sg]

thank you :)

- on my other post, i had a question written down, and royalcat, you said i had to use vector addition again. the question was"The driver of a motor boat points it directly toward the opposite bank of a 52 m wide river. The speed of the boat is 4.0 m/s and the river flows at 3.2 m/s. When the boat reaches the opposite riverbank, what is the distance downstream from its point of departure? "



could i divide 52 by 4, because that's how many seconds it takes to go down the river, then times it by the speed of the river( 3.2) to find the distance?
or is that just completely wrong...

 

[RoyalCat]

thank you :)

- on my other post, i had a question written down, and royalcat, you said i had to use vector addition again. the question was"The driver of a motor boat points it directly toward the opposite bank of a 52 m wide river. The speed of the boat is 4.0 m/s and the river flows at 3.2 m/s. When the boat reaches the opposite riverbank, what is the distance downstream from its point of departure? "
could i divide 52 by 4, because that's how many seconds it takes to go down the river, then times it by the speed of the river( 3.2) to find the distance?
or is that just completely wrong...

I jumped the gun a bit at the vector addition. Since the two velocities are orthogonal, you don't need to add the vectors since the velocity of the river doesn't affect how long the crossing takes.

Let's define our y-axis as the north-south direction, and the x-axis as the east-west direction.

$$d_y=vt_y$$
$$t_y$$ is our unknown.
$$t=d/v$$
$$t_{crossing}=\frac {52 m}{4 m/s} = 13 sec$$

$$d_x=vt$$
$$d_x$$ is our unknown this time, since we're looking for $$d_x$$ at the moment $$t_{crossing}$$
$$d_x= 3.2 m/s * 13 sec = 41.6 m$$