00:01
So in this problem, we're given this boat on this river, and the river is flowing east at 10 miles per hour.
00:14
So east would be this way because north is up, according to the picture we've been given.
00:20
And the boulder starts at the south shore and heads in the direction 60 degrees from the shore by this picture.
00:29
And the motorboat has a speed of 20 miles per hour.
00:38
20 miles per hour relative to the water okay so first we're asked to express the velocity of the river as a vector in component form so what does that mean? well since if we would draw this all in coordinate plane right we would draw it like this this will be x, this would be y.
01:44
And the river vector would be like such, right? i'll call it u here, vector u.
01:58
Okay, and so then u would be the vector 10 cosine of theta.
02:13
Well, theta here is zero degrees, isn't it? and then 10 sine 0.
02:22
Well, the sign of 0 is 0.
02:26
So this is the vector, and the cosine of 0 is 1.
02:30
So this is the vector 10, 0.
02:34
Okay.
02:36
Now, next, rest express the velocity of the motorboat relative to the water as a vector in component form.
03:03
Okay.
03:04
So, again, if we think of this on our coordinates here, right, we would have this vector here.
03:18
I'll call it v, where this theta is 60 degrees now.
03:25
Okay.
03:26
And we're doing 20 miles an hour relative to the water.
03:31
And so the vector v is 20 cosine of theta, which is 60 degrees, and then 20 sine.
03:47
Of 60 degrees well cosine of 60 degrees is a half so 20 times a half that's 10 cos the sign of 60 degrees this is a half and this is the square of 2 over 2 i'm sorry square of 3 let me fix that i'm at square of 3 3 square of 2 or 2 is 45 this is 60 okay so this is 10 of the square of a square of a square of 3 three all right next rest to find the true velocity of the motor boat true velocity of motor boat well true velocity would simply be these two vectors added together so that's the vector u plus the vector v in it which is 10 zero plus 10, 10 times square of 3.
05:22
To add vectors, you add them component -wise...