Linear Algebra questions :S??

Question

Plz lemme no if u no the answers 2 any or all these questions

1) Let a = (1, 8, 1) and b = (3, 3, 4) be vectors.
Compute the cross product aXb. (?,? ,?)

2) If a = i + 10j + k and b = i + 19j + k, find a unit vector with positive first coordinate orthogonal to both a and b.
_i + _j + _k

3)Find a unit vector with positive first coordinate that is orthogonal to the plane through the points P = (5, 2, 0), Q = (9, 6, 4), and R = (9, 6, 7).
(?,?,?)

4)Find the parametric equations for the line through the point P = (-2, 3, 0) that is perpendicular to the plane 1x - 5y + 0z = 1.
Use "t" as your variable, t = 0 should correspond to P, and the velocity vector of the line should be the same as the standard normal vector of the plane. x= y= z=

At what point Q does this line intersect the yz-plane? Q=(?,?,?)

5)) Consider the two lines
L1: x = -2t, y = 1 + 2t, z = 3t and
L2: x = -8 + 4s,y = 0 + 5s, z = 5 + 1s

Find the point of intersection of the two lines.
P = (, , )

Answer 1

One problem per question please.

3) Find a unit vector with positive first coordinate that is orthogonal to the plane through the points P = (5, 2, 0),
Q = (9, 6, 4), and R = (9, 6, 7).

First use the points in the plane to define two directional vectors u and v, of the plane.

u = PQ = = <9-5, 6-2, 4-0> = <4, 4, 4>
v = PR = = <9-5, 6-2, 7-0> = <4, 4, 7>

The normal vector n, of the plane is orthogonal to every vector that lies in the plane. Take the cross product.

n = u X v = <4, 4, 4> X <4, 4, 7> = <12, -12, 0>

Any non-zero multiple of n is also a normal vector of the plane. Divide by 12.

n = <1, -1, 0>

Calculate the magnitude of n.

|| n || = √[1² + (-1)² + 0²] = √(1 + 1 + 0) = √2

To make n a unit vector, divide by its magnitude.

n / || n || = n = <1, -1, 0> / √2

A unit normal vector of the plane is:

<1/√2, -1/√2, 0>