Verifying Vector Intersection: Two Questions

In summary, the conversation discusses two questions regarding vectors and determining if they are inclined at a certain angle or perpendicular. The first question involves finding the values of a for which the vectors i+3j-k and i+aj+k meet certain conditions, while the second question involves proving that two lines intersect. The conversation also includes a discussion on how to find the point of intersection for the lines.
  • #1
Fila
5
0
I have two questions I need to make sure if I'm doing correctly. Its vectors.


1)
For what values of a are the vectors i+3j-k and i+aj+k

i) inclined at 30 degree angle
cos@ = n1.n2 / |n1||n2|
cos^2(30) * 11(2+a^2) = ((3a)^2)
a=sqrt22


ii) perpendicular
(1,3,-1).(1,a,1)=0
1+3a-1=0
a=0


2)
Show that the lines
r=s(i+2j+3k)
r=(3i+5j+4k)+t(2i+3j+k)
intersect

I believe I should try and prove that they are not parallel. If their dot product is zero, then it means the lines are parallel. So if the answer is not zero, then I proved that the lines do intersect.
Do I just take the cross product of (1,2,3) and (2,3,1)? I want to know if I'm using the correct vectors to do the dot product.
 
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  • #2
1) is correct.

Your reasoning for 2) is not correct. Two lines in 3 dimensions which are not parallel do not necessarily intersect. Instead, simply try working out the point of intersection.
 
  • #3
Thanks!

How do I get started with finding the point of intersection in #2? I couldn't find this 3 dimension problem in the book.
 
Last edited:
  • #4
I'll try to help get you started. You have two vector equations and two parameters. You can set up three equations, each of which have the same two unknowns, by equating the x, y and z components of each vector equation. You equate x,y and z(or i,j and k depending on how you wish to state it) components of the two vector equations because at the POI they are equal.

i: s = 3 + 2t (1)
j: 2s = 5 + 3t (2)
k: 3s = 4 + t (3)

Solve any two of the three equations simultaneously to obtain values for t and s. So for example solving (1) and (2) simultaneously will yield: s = 1 and t = -1. Does (s,t) = (1,-1) satisify equation 3? If they do not satisfy equation 3 then the lines cannot intersect. If they do satisfy equation 3 then I'll leave it up to you to figure out what that means.
 
  • #5
I had no idea how to write the two lines into those 3 equations to solve it. Now I know what to do. Thanks!
 

1. What is vector intersection and why is it important in science?

Vector intersection refers to the point where two or more vectors intersect or cross each other. In science, it is important because it helps us understand the relationship between multiple vectors and their impact on a particular system or phenomenon.

2. How do you verify vector intersection?

The most common method to verify vector intersection is by using mathematical calculations. This involves finding the equations of the vectors and solving them simultaneously to determine the coordinates of the intersection point. Other methods may include graphical representation or physical experimentation.

3. What are some real-life applications of verifying vector intersection?

Verifying vector intersection has numerous real-life applications, such as in navigation systems, where it is used to determine the intersection point of multiple paths or routes. It is also used in engineering and construction to analyze the stability and strength of structures, and in physics to study the motion of objects.

4. Can vector intersection be verified in three-dimensional space?

Yes, vector intersection can be verified in three-dimensional space using the same methods as in two-dimensional space. However, the calculations become more complex and may require the use of advanced mathematical techniques.

5. What are some challenges in verifying vector intersection?

One of the main challenges in verifying vector intersection is ensuring the accuracy of the calculations, as even a small error can significantly affect the results. Another challenge is dealing with non-linear vectors or vectors with changing magnitudes, which may require more advanced methods to verify their intersection.

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