# Thread: Vector equation of a line

1. ## Vector equation of a line

I can not find a solution to this question at all, anyone any ideas?

Find a unit vector that is perpendicular to both 4i + 10j - 3k and 5i - 9j + 7k

I am as far as finding the vector between them, i - 19j + 10k

I have called the new vector n1 i + n2 j + n3 k

Since they are going to be perpendicular n1 -19n2 +10n3 = 0

I am now stumped.

2. If each of $U~\&~V$ are vectors:
the vector $\frac{{U \times V}}{{\left\| {U \times V} \right\|}}$ is a unit vectot perpendicular to both;

$\arccos \left( {\frac{{U \cdot V}}{{\left\| U \right\|\left\| V \right\|}}} \right)$ is the angle between them.

3. I have been told that this method is not on my specification, are there any others?

4. Originally Posted by r4mbini
I can not find a solution to this question at all, anyone any ideas?

Find a unit vector that is perpendicular to both 4i + 10j - 3k and 5i - 9j + 7k

I am as far as finding the vector between them, i - 19j + 10k

I have called the new vector n1 i + n2 j + n3 k

Since they are going to be perpendicular n1 -19n2 +10n3 = 0

I am now stumped.

Here is what you need to do:

$(n_1 i + n_2 j + n_3 k) \cdot (5i - 9j + 7k) = 0 \Rightarrow 5 n_1 - 9 n_2 + 7 n_3 = 0$ .... (1)

$(n_1 i + n_2 j + n_3 k) \cdot (4i + 10j - 3k) = 0 \Rightarrow 4n_1 + 10n_2 - 3n_3 = 0$ .... (2)

$n_1^2 + n_2^2 + n_3^2 = 1$ .... (3)

Solve equations (1), (2) and (3) simultaneously.

5. I now feel very foolish :P