Vectors

[chris12121]

Need help with this question - please show working thanks

The line L1 passes through the point (3,6,1) and is parallel to the vector 2i+3j-k. The line L2 passes through the point (3,-1,4) and is parallel to the vector i-2j+k

i) Write down vector equations for the lines L1 and L2
ii) Calculate the acute angle between the lines

 

[mmm4444bot]

?
Hi Chris:

Since you did not ask any questions of your own about vector equations or angles between vectors, I'm not sure where you're stuck.

You've also made no statements regarding what you already know about vectors, so it's difficult to know where to begin helping you.

I'll just give an example. If you need more help, then please try to be more specific.



Write the vector equation of the line passing through point P with direction vector v.

P = (2, -1, 3)

v = <-3, 4, 2>

This line is composed of all points Q such that Q = P + t * v

where t is any Real number.

So we write:

(x, y, z) = (2, -1, 3) + t * <-3, 4, 2>

That is the vector equation of the line.

The points (x, y, z) on this line can also be written out in terms of their coordinates, using separate equations. When we do this, we call the following equations the "parametric equations for the line".

x = 2 - 3t

y = -1 + 4t

z = 3 + 2t



The angle ? between two non-zero vectors can be found from the following relationship involving the cosine of ?, the dot-product, and the norm (that is, magnitude, or length) of each vector.

The symbolism for a vector norm varies from author to author.

|v| is used to denote the norm; so is ||v||.

If you're not familiar with the dot-product, you should be able to find a reference in your textbook's index.

\(\displaystyle cos(\theta) \ = \ \frac{\vec{u} \cdot \vec{u}}{||\vec{u}|| \ ||\vec{v}||}\)



Give it a try. If you would like more help, please show us what you've done so far, or tell us what you're thinking.

Cheers ~ Mark