1. ## Vector line equations

I've done the first part of the attached question. 6lambda:1mu.

I don't think the hints make sense. c is perpendicular to a. How can it be on a plane with a and b and also be parallel to l as implied by r=a+kc?

2. c is not parallel to l. r = a + kc means that each point of l is a resultant of a and some vector collinear with c. This is guaranteed becase a and c form a basis for the plane that l lies in.
This is akin to the vector equation of a line with standard basis: r = b + kx.

3. Originally Posted by Stuck Man
I don't think the hints make sense. c is perpendicular to a. How can it be on a plane with a and b and also be parallel to l as implied by r=a+kc?
The vector $a\times(a\times b)=(a\cdot b)a-(a\cdot a)b$ is clearly perpendicular to $a$ and as a linear combination of $a~\&~b$ it is in the plane of $a~,~b,~&~O$.
Now you can finish?

4. Here is what you cannot read
The vector a x (a x b)=(a.b)a-(a.a)b is clearly perpendicular to a and as a linear combination of $a~\&~b$ it is in the plane of $a~,~b,~\&~O$.

5. Here is what you cannot read
The vector a x (a x b)=(a.b)a-(a.a)b is clearly perpendicular to a and as a linear combination of $a~\&~b$ it is in the plane of a, b, & O.

6. Originally Posted by slider142
c is not parallel to l. r = a + kc means that each point of l is a resultant of a and some vector collinear with c.
My book says "The equation of a straight line parallel to vector b through a point with position vector a is r=a+tb." That is why I thought the line l would be parallel to c.

7. I have consulted another source which was helpful and can confirm that you are wrong slider142.

I had been thinking that c is a position vector so I can understand it all better now. slider142 also seems to have thought that c is a position vector.

I calculated that lambda:mu is 6:1 but the book says 1:6. Is the book wrong? I think I can do all the rest of the question.

8. After 3 days still no one has given me any help.

Why is lambda:mu 1:6? Is the equation of line l r=i-2j+k+t(13i+4j-5k) as the book says? Why is p=-12i-6j+6k? Why is p not equal to b?

9. I've finally done the question. Line l actually meets OB produced.