Linear independence and vector parametrics

[hadroneater]

Homework Statement

Consider the two lines, given in the paramentric form

L1: x = (0, 1 ,2) + s(1, 0, 2)
L2: x = (4, 2, c) + t(-2, 0, d)

where c and d are constants.

a) For what value of d are the lines parallel?
b) With the value of d above, for what value(s) of c (if any) are the two lines identical? Justify briefly.
c) For the case c = 5 and d = 0 find the point P on L1 and Q on L2 such that the distance between P and Q is as small as possible.

2. A concept question:
Can any vector x with two components be expressed as a linear combination of a and b? Why?

Homework Equations

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The Attempt at a Solution

a) I think d = -4, but not sure if that's right.
b) I'm not sure how to approach this problem. Do I make 2 + 2s = c - 4t
But what do I do after that?
c) No clue. I'm thinking perhaps using the dot product = 0?

2. I think a vector can only be expressed as a linear combination of a and b if the two vectors are linearly independent. How do I prove that?

 

[mighty2000]

Thank you for posting your questions. I am a scientist and I would be happy to help you with your concerns.

For part a), your answer of d = -4 is correct. To find this value, you can equate the z-coefficients of the two lines, which gives you the equation 2 = c - 4t. Then, since the lines are parallel, their direction vectors (1, 0, 2) and (-2, 0, d) must be parallel as well. This means their cross product must be equal to 0, which leads to the equation 2d = 0. Solving for d gives you d = -4.

For part b), you can substitute the value of d = -4 into the equation you obtained in part a). This gives you c = 2 + 2s. Since s is a parameter, the lines will be identical for any value of c that satisfies this equation. Therefore, the two lines will be identical for all values of c = 2 + 2s, where s is a real number.

For part c), you are correct in thinking that the dot product can help you find the point P and Q with the smallest distance. The dot product of two vectors is related to the angle between them, and when the angle is 0, the distance between the two points is minimized. You can use this fact to set up an equation for the dot product of the direction vectors of the two lines, and then solve for the values of s and t that minimize the distance between the two points. Once you have the values of s and t, you can substitute them into the equations of the two lines to find the points P and Q.

For the concept question, you are correct in thinking that a vector can only be expressed as a linear combination of two vectors if they are linearly independent. This means that one vector cannot be a multiple of the other, and they must have different directions. This can be proven by setting up a system of equations and solving for the coefficients of the linear combination, showing that there is no unique solution when the vectors are linearly dependent.

I hope this helps clarify your understanding of these concepts. Please let me know if you have any further questions.