A canoe is moving with velocity vector v==5i+3j m/sec relative to the water. The velocity of the current in the water is the vector c==i+2j m/sec. What is the speed of the boat relative to an observer standing on land? What is the speed of the current in the direction of the canoe's motion relative to the water?

Question

A canoe is moving with velocity vector v=<5,3>=5i+3j m/sec relative to the water. The velocity of the current in the water is the vector c=<1,2>=i+2j m/sec. What is the speed of the boat relative to an observer standing on land? What is the speed of the current in the direction of the canoe's motion relative to the water?

anonymous · Answer

I'm told that answer to the first part is square root of 61, why? and the second part answer is (11/sqrt(34)) why?

rulnick · Answer

sum vector is <6,5> which has "length" (speed) sqrt ( 6^2 + 5^2) = sqrt(61).

rulnick · Answer

speed of current in frame of reference of canoe is (1(5)+2(3)) / sqrt(5^2+3^2)
 = 11/sqrt(34)

rulnick · Answer

okay?

anonymous · Answer

but why am I summing the vectors v+c I drew it...and the second part why is the dot product v.c being taken in the numerator?

rulnick · Answer

Summing because both contribute to motion relative to land.
Dot product because that's precisely how to account for frame of reference; dot with the normalized vectors of the reference frame (normalized by dividing by length).

anonymous · Answer

Perfect for the response "summing both contribute to motion relative to land"!!! I will put more thought on the second part since I know dot products are intimately related to lengths and angles. Thank you!!! God bless :-)