Answer:
(A) 120 mph
(B) -120 mph
(C) 12 hours 5 minutes.
Step-by-step explanation:
Given that,
New York and California are 2900 miles apart,
Both the trains A as well as B travel at an average speed of 120 mph.
Assuming that the path between the starting point in New York and the ending point in California is straight, so the given distance of 2900 miles is actually the shortest distance which is the displacement between the source and destination.
So, the given speed on the straight path is actually the velocity.
Assuming that the velocity is positive in the direction from New York towards California.
(A) So, the velocity of train A (towards California) = 120 mph.
(B) The velocity of train B (towards New York) = -120 mph.
(C) Bothy are moving towards each other, so the relative velocity between them is 120+120=240 mph, and the initial distance between them is 2900 miles.
When they meet, the displacement between them becomes zero, displacement covered= 2900-0=2900 miles.
So, the time, t, taken to cover the displacement 2900 miles with a relative velocity of 240 mph is
[as time= displacement / relative velocity]
hours
=12 hours 5 minutes.
Answer:
a. Line
b. Plane
c. All of R^3
Step-by-step explanation:
In order to answer this question, we need to study the linear independence between the vectors :
1 - A set of three linearly independent vectors in R^3 generates R^3.
2 - A set of two linearly independent vectors in R^3 generates a plane.
3 - A set of one vector in R^3 generates a line.
The next step to answer this question is to analyze the independence between the vectors of each set. We can do this by putting the vectors into the row of a R^(3x3) matrix. Then, by working out with the matrix we will find how many linearly independent vectors the set has :
a. Let's put the vectors into the rows of a matrix :
⇒ Applying matrix operations we find that the matrix is equivalent to this another matrix ⇒
We find that the second vector is a linear combination from the first and the third one (in fact, the second vector is the first vector multiply by -3).
We also find that the third vector is a linear combination from the first and the second one (in fact, the third vector is the first vector multiply by 5).
At the end, we only have one vector in R^3 ⇒ The set of all linear combinations of the set a. is a line in R^3.
b. Again, let's put the vectors into the rows of a matrix :
⇒ Applying matrix operations we find that the matrix is equivalent to this another matrix ⇒
We find that there are only two linearly independent vectors in the set so the set of all linear combinations of the set b. is a plane (in fact, the third vector is equivalent to the first vector plus three times the second vector).
c. Finally :
⇒ Applying matrix operations we find that the matrix is equivalent to this another matrix ⇒
The set is linearly independent so the set of all linear combination of the set c. is all of R^3.