I need help /w equations of some lines

**Mike49214** · July 31st 2008, 03:46 PM

Hey everyone!

I can't seem to solve these two questions. Any help is appreciated!

1) A line has direction angles of 60 degrees, 45 degrees and 60 degrees. It passes through the point (1, -2, 5). Determine the vector equation of this line.

2) Find the vector equation of a line that intersects the following 2 lines at 90 degrees.
Line 1: (x,y,z) = (4, 8, -1) + a(2, 3, -4)
Line 2: (x, y,z) = (7, 2, -1) + a(6, 1, 2)

Thanks in advance
-Mike

**earboth** · August 1st 2008, 04:59 AM

Originally Posted by Mike49214

...

2) Find the vector equation of a line that intersects the following 2 lines at 90 degrees.
Line 1: (x,y,z) = (4, 8, -1) + a(2, 3, -4)
Line 2: (x, y,z) = (7, 2, -1) + a(6, 1, 2)

Thanks in advance
-Mike

1. The direction vectors of the lines are different therefore the lines are not parallel.
2. An attempt to calculate the point of intersection doesn't yield a result therefore the lines are skewed.
3. The cross-product of the direction vectors yield the direction perpendicular to both lines:

$(2,3,-4) \times (6,1,2) = (10, -28, -16)$

4. Now use this direction vector and any arbitrary point on $L_1$ to get the equation of a family of parallel lines:

$p: (x,y,z) = q \cdot (10,-28,-16)+\left((4, 8, -1) + r(2, 3, -4)\right)$ ....... r is considered to be constant with a certain line of the family.

5. One (and only one) member of this family will intersect the line $L_2$ :

$q \cdot (10,-28,-16)+\left((4, 8, -1) + r(2, 3, -4)\right) = (7, 2, -1) + s(6, 1, 2)$

6. Solve this system of linear equations for r, s and q.
I've got $q=-\frac{33}{190}~\wedge~s=-\frac{48}{95}~\wedge ~=-\frac{36}{95}$

7. The 2 points on different lines which have the smallest distance are:

$P\left(\frac{449}{95}~,~\frac{154}{95}~,~ -\frac{167}{95}\right)$ and $Q\left(\frac{284}{95}~,~\frac{616}{95}~,~ \frac{97}{95}\right)$

8. Take one of the two points to calculate the equation of the line:

$p: (x,y,z) = \left(\frac{449}{95}~,~\frac{154}{95}~,~ -\frac{167}{95}\right) + t \cdot (10,-28,-16)$

My results do look really ugly, so better check my arithmetic. (If they are miraculously correct you are allowed to think two or three ugly words about the author of this question)

**Mike49214** · August 2nd 2008, 10:10 AM

Originally Posted by earboth

...

I've gone over it like 3 times and I can't see any errors in your calculations. Thanks a lot for the help, I appreciate it.

I am still having trouble with the first question though, if anyone would be nice enough to lend a hand.

**earboth** · August 2nd 2008, 11:31 AM

Originally Posted by Mike49214

I've gone over it like 3 times and I can't see any errors in your calculations. Thanks a lot for the help, I appreciate it.

I am still having trouble with the first question though, if anyone would be nice enough to lend a hand.

Nobody of the forum has answered your question because it isn't understandable:

You mention three angles. Which geometric objects form the angles? The line and axes? (if yes which axis belongs to which angle?), planes? (if yes which plane belongs to which angle?)

... and if you could provide a rough sketch that would really help us to help you.

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