Math Help - Vector help

1. Vector help

GCE Maths
With respect to an origin O, the pts. A and B have the following position vectors

OA=2i+2j+k OB=i+4j+3k

The line has vector equation
r=4i-2j+2k+s(i+2j+k)

I. Prove that the line l does not intersect the line through A and B

II. Find the equation of the plane containing l and the point A giving your answer in the form ax+by+cz=d

Originally Posted by I-Think
GCE Maths
With respect to an origin O, the pts. A and B have the following position vectors

OA=2i+2j+k OB=i+4j+3k

The line has vector equation
r=4i-2j+2k+s(i+2j+k) what is s?
is this "line" a sum of the vectors OA;OB? if so is the point of the exercise to find s?

I. Prove that the line l does not intersect the line through A and B
If a and b are vectors it does not matter where on the cordinate system you place them. Not sure what you mean here...?

II. Find the equation of the plane containing l $\Rightarrow$ what is this? and the point A giving your answer in the form ax+by+cz=d
$\Rightarrow$ is that the sum of vectors OA;OB?
Regards H (sorry but really difficult to understand without a picture)..

3. Hello I-Think
Originally Posted by I-Think
GCE Maths
With respect to an origin O, the pts. A and B have the following position vectors

OA=2i+2j+k OB=i+4j+3k

The line has vector equation
r=4i-2j+2k+s(i+2j+k)

I. Prove that the line l does not intersect the line through A and B

II. Find the equation of the plane containing l and the point A giving your answer in the form ax+by+cz=d
First we need to find the vector equation of the line through $A$ and $B$. This can be written as
$\vec r = \vec{OA}+t\vec{AB}$ ...(1)
where $t$ is a scalar parameter. Do you understand this bit? It simply says:
$\underbrace{\vec r=}_{\text{You can move from O to any point on the line ... }}$ $\underbrace{\vec{OA}}_{\text{ by a movement to A ...}}\underbrace{+}_{\text{ followed by ...}}\underbrace{t\vec {AB}}_{\text{ a movement in the direction of AB}}$
Now
$\vec{OA} = 2\vec i + 2\vec j + \vec k$
and
$\vec{AB} = \vec{OB} - \vec{OA}$
$= -\vec i +2 \vec j +2\vec k$
So, from (1), the equation of the line through $AB$ is:
$\vec r = 2\vec i + 2\vec j + \vec k+t(-\vec i +2 \vec j +2\vec k)$
Next, we need to prove that this line doesn't intersect line $l$. We do this by showing that there are no values of $s$ and $t$ that will bring us to the same point on each of the lines. So, suppose that the lines meet. Then, for some value of $s$ and of $t$:
$2\vec i + 2\vec j + \vec k+t(-\vec i +2 \vec j +2\vec k)= 4\vec i-2\vec j+2\vec k+s(\vec i+2\vec j+\vec k)$
Comparing the coefficients of $\vec i$ and $\vec j$, we get:
$2-t = 4+s$
and
$2+2t = -2+2s$
which gives:
$s = 0,\;t=-2$
If we now compare the coefficients of $\vec k$:
On the LHS: $1-4 = -3$

On the RHS: $2 + 0 = 2$
Since these are not equal, there are no values of $s$ and $t$ that satisfy the equation, and hence the lines do not intersect.

Part II

We have a choice of two methods here. The first uses simple algebra; the second, rather more sophisticated, uses a vector and scalar product.

For the first method, we find a couple of points on the line $l$ by taking two values of $s$, and then solve a set of simultaneous equations.

So, with $s = 0$ and $s = 1$, we get the points $(4, -2, 2)$ and $(5, 0, 3)$ lying on $l$, and therefore also in the plane. And the point $A$ also lies in the plane.

So, if the plane is $ax+by+cz=1$ (which is simply the equation $ax+by+cz=d$ re-written when divided throughout by $d$, using new values of $a, b, c$), we get:
$\left \{\begin{array}{l}
4a-2b+2c=1\\
5a+0b+3c=1\\
2a+2b+c=1\\
\end{array}\right.$

This gives:
$\left \{\begin{array}{l}
a=1\\
b=\tfrac16\\
c=-\tfrac43\\
\end{array}\right.$

and, when we get rid of fractions, gives the plane:
$6x+y-8z=6$
If you want to see an explanation of the vector method, look at this page.