# A brief vector question

• Mar 1st 2009, 12:23 AM
JeWiSh
A brief vector question
This vector question has three parts, and I've done the first two, finding the point of intersection and acute angle between them. It's the third bit I don't understand.

L1 = 2i-j+3k + s(j-k)
L2 = -2i+7j+3k + t(2i-3j-k)

iii) The foot of the perpendicular from the point with position vector -2i+7j+3k to the line L1 is P. Find the position vector of P.

I just don't really understand what this means, any help would be much appreciated.
• Mar 1st 2009, 07:08 AM
earboth
Quote:

Originally Posted by JeWiSh
This vector question has three parts, and I've done the first two, finding the point of intersection and acute angle between them. It's the third bit I don't understand.

L1 = 2i-j+3k + s(j-k)
L2 = -2i+7j+3k + t(2i-3j-k)

iii) The foot of the perpendicular from the point with position vector -2i+7j+3k to the line L1 is P. Find the position vector of P.

I just don't really understand what this means, any help would be much appreciated.

1. Let P denote the foot of the perpendicular line to L1 and Q(-2, 7, 3). Then PQ is the shortest distance between the line and Q.

2. Calculate the distance between Q and any arbitrary point on L1:

$\displaystyle d(s) = \sqrt{\left((2, -1, 3) + s\cdot (0, 1, -1) - (-2, -7, 3)\right)^2} = \sqrt{\left((4, -8, 0)+s\cdot(0, 1, -1)\right)^2}$

3. Let $\displaystyle D(s) = (d(s))^2$. Calculate now the minimum of D:

$\displaystyle D(s)=2s^2 - 16s + 80~\implies~ D'(s)=4s-16$

Therefore s = 4

4. Plug in this value into the equation of L1 to calculate the coordinates of P:

$\displaystyle \vec F = 2i-j+3k + 4 \cdot (j-k) = (2i + 3j - 1k)$
• Mar 1st 2009, 07:12 AM
Soroban
Quote:

This vector question has three parts, and I've done the first two,
finding the point of intersection and acute angle between them.
It's the third bit I don't understand.

$\displaystyle L_1\!:\;\;2i-j+3k + s(j-k)$

iii) The foot of the perpendicular from the point $\displaystyle A$
with position vector: $\displaystyle -2i+7j+3k$ to line $\displaystyle L_1$ is $\displaystyle P.$
Find the position vector of $\displaystyle P.$

Code:

          A             *             \               \               \              L1                 \          *                 \    *                   *             *    P         *

$\displaystyle AP$ is drawn perpendicular to $\displaystyle L_1$

We are asked to locate point $\displaystyle P.$