Equivalent Vectors...3

[nycmathguy]

Section 6.3

 

[MathLover1]

correct

 

[nycmathguy]

correct

I feel like traveling on into math land.

 

[Country Boy]

NO! It is not correct! You are comparing the lengths of the vectors but having the same length is not enough. For example, the vector from (0, 0) to (4, 0) and the vector from (0, 0) to (0, 4) have the same length but are NOT "equivalent" because they do not have the same direction.

One way to show that two vectors are equivalent is to show that they have the same length and the same direction.
For example, in problem 18, u has initial point (8, 1) and terminal point (13, -1). It has lenght $\sqrt{(8- 13)^2+ (1- (-1))^2}= \sqrt{5^2+ 2^2}= \sqrt{29}$ as you say. It also has direction arctan(2/5) degrees. v has initial point (-2, 4) and terminal point (-7, 6). Its length is $\sqrt{(-2- (-7))^2+ (4- 2)^2}= \sqrt{5^2+ 2^2}=\sqrt{29}$ as before. The two vectors are the same length but are they in the same direction? The direction of v is arctan(2/5) as before so yes, they are in the same direction, so are equivalent vectors, but your reasoning is incorrect because you did not check the directions.

 

[nycmathguy]

NO! It is not correct! You are comparing the lengths of the vectors but having the same length is not enough. For example, the vector from (0, 0) to (4, 0) and the vector from (0, 0) to (0, 4) have the same length but are NOT "equivalent" because they do not have the same direction.

One way to show that two vectors are equivalent is to show that they have the same length and the same direction.
For example, in problem 18, u has initial point (8, 1) and terminal point (13, -1). It has lenght $\sqrt{(8- 13)^2+ (1- (-1))^2}= \sqrt{5^2+ 2^2}= \sqrt{29}$ as you say. It also has direction arctan(2/5) degrees. v has initial point (-2, 4) and terminal point (-7, 6). Its length is $\sqrt{(-2- (-7))^2+ (4- 2)^2}= \sqrt{5^2+ 2^2}=\sqrt{29}$ as before. The two vectors are the same length but are they in the same direction? The direction of v is arctan(2/5) as before so yes, they are in the same direction, so are equivalent vectors, but your reasoning is incorrect because you did not check the directions.

Are you saying that MathLover1 is wrong? She's pretty good at solving math problems.