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I am completely lost with this question. I have no idea where to start or what to do exactly. Any help would be greatly appreciated.

(a) Assume vector u_1 and vector u_2 are two vectors of the same length. Show that vector w = vector u_1 + vector u_2 bisects
the angle between vector u_1 and vector u_2. Sketch the situation.

(b) Consider the planes pi_ 1 : 4y + 3z = -4 and pi_ 2 : x - 2y + 2z = 2.

i. Show that these planes intersect in a line and nd the angle between them.
ii. Find an equation for the plane pi_ 3 that bisects the acute angle between pi_1 and pi_ 2.
iii. To check your work in (b), find the angle between pi_ 3 and pi_1.

Answer
Questioner:   Nick
Country:   Quebec, Canada
Category:   Advanced Math
Private:   No
Subject:   linear algebra

I am completely lost with this question. I have no idea where to start or what to do exactly. Any help would be greatly appreciated.

(a) Assume vector u_1 and vector u_2 are two vectors of the same length. Show that vector w = vector u_1 + vector u_2 bisects
the angle between vector u_1 and vector u_2. Sketch the situation.

(b) Consider the planes pi_ 1 : 4y + 3z = -4 and pi_ 2 : x - 2y + 2z = 2.

i. Show that these planes intersect in a line and nd the angle between them.
ii. Find an equation for the plane pi_ 3 that bisects the acute angle between pi_1 and pi_ 2.
iii. To check your work in (b), find the angle between pi_ 3 and pi_1.
.........................................
(a) Remember your HS geometry class?  Look up the theorems about a RHOMBUS, such as:

The diagonals of a rhombus bisect the angles.

(b) 1. the angle between the planes is the same as the angle between their normal vectors.  Look up the formula for the angle between two vectors. (Actually the cosine of ...)

The line of intersection can be found by solving the system of equations (for  pi-1 and pi-2; and treating the solution parametrically.  (That means, pick  t = one of x,y,z, then solve for the others in terms of t)


2. Use part (a) -- add the normal vectors (as unit vectors, of course, so their lengths will be equal.), then find some point in the two planes.

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