Finding the Point of Intersection and Acute Angle between Two Lines

In summary, the task is to prove the intersection of two lines, r=(a,b,c)+t(d,e,f) and q=(h,i,j)+s(k,l,m), and find the coordinates of their point of intersection. To find the acute angle between the lines, the equations x= a+ td= h+ sk, y= b+ te= i+ sl, z= c+ tf= j+ sm can be used to solve for t and s. Typically, two lines in 3 dimensions do not intersect.
  • #1
nk735
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Homework Statement



Prove that the lines r=(a,b,c)+t(d,e,f) and q=(h,i,j)+s(k,l,m) intersect, and find the coordinates of their point of intersection. Also, find the acute angle between their lines

Homework Equations





The Attempt at a Solution



I have no attempt because I'm stumpped...
 
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  • #2
Do you understand that "r= (a,b,c)+ t(d,e,f)" means that x= a+ td, y= b+ te, z= c+ tf for every point on that line and that "q= (h,i,j)+ s(k,l,m)" means that x= h+ sk, y= i+ sl, z= j+ sm for every point on THAT line. If the two lines intersect then, at the point of intersection they have the same x, y, z values: x= a+ td= h+ sk, y= b+ te= i+ sl, z= c+ tf= j+ sm. That gives you three equations to solve for t and s.

Of course, typically, you can use two of those equations to find t and s and the check in the third to see if they work. Typically, two lines in 3 dimensions do NOT intersect.
 
  • #3
Ok, I've managed to get the point of intersection from that, thankyou. But what about the acute angle between them?
 

1. What is the intersection of two lines?

The intersection of two lines is the point where the two lines meet or cross each other. It is the solution to the equations of the two lines and represents the coordinates where they intersect.

2. How do you find the intersection of two lines?

To find the intersection of two lines, you need to solve the equations of the two lines simultaneously. This can be done by setting the equations equal to each other and solving for the variables. The resulting values will be the coordinates of the intersection point.

3. Can two lines intersect at more than one point?

No, two lines can only intersect at one point. This is because two lines can only have one solution when solving their equations simultaneously. If two lines intersect at more than one point, they are not considered to be distinct lines, but rather the same line.

4. What is the significance of the slope of two intersecting lines?

The slope of two intersecting lines is significant because it represents the rate of change or steepness of the lines. The slope of the lines must be different in order for them to intersect at a single point. If the slopes are the same, the lines are parallel and will never intersect.

5. Can two intersecting lines have the same y-intercept?

Yes, two intersecting lines can have the same y-intercept. This means that they will intersect at a point where the x-coordinate is the same for both lines, but the y-coordinate will be different. In this case, the lines are not parallel and will intersect at a single point.

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