a) find the vector equations of the two lines that bisect the angles between the lines.....?

Question

a) find the vector equations of the two lines that bisect the angles between the lines:

r1= [5,2] + t[-3,6] and r2 = [5,2] + s[11,2]

b) are the two lines that bisect the angles made by the intersecting lines always perpendicular? Explain.

Answer 1

Given the two lines

r1= [5,2] + t[-3,6]
r2 = [5,2] + s[11,2]

If we set s and t equal to zero, we see they both have a point (5,2) in common, the point of intersection of the lines.

To find one of the directional vectors v1, of a line that bisects the given lines, add the unit vectors from both given lines.

Let
u1 = [-3,6]
u2 = [11,2]

|| u1 || = √[(-3)² + 6²] = √(9 + 36) = √45 = 3√5
|| u2 || = √[11² + 2²] = √(121 + 4) = √125 = 5√5

v1 = u1 / || u1|| + u2 / || u2 || = [-3,6]/(3√5) + [11,2]/(5√5)
v1 = [5*(-3) + 3*11, 5*6 + 3*2] / (15√5)
v1 = [18, 36] / (15√5) = [6, 12] / (5√5)

Any non-zero multiple of the vector will also be a directional vector. Multiply by (5√5) / 6.

v1 = [1, 2]

So the equation of the first line that bisects the intersection of the two given lines is:

L1 = [5,2] + s[1, 2]

The second line also goes thru the same point (5,2) at right angles to the first line. So the dot product of the two directional vectors is zero.

L2 = [5,2] + t[2, -1]

As a check, take the dot product of the directional vectors of L1 and L2.

dot product = [1, 2] • [2, -1] = 2 - 2 = 0

So this checks.

Answer 2

a)

As luck would have it, these lines intersect at [5, 2]. A direction vector of one of their angular bisectors is given by the mean of the unit direction vectors of both lines:

([-3,6]/(5√3) + [11, 2]/5√5) / 2 = [-3√5 +11√3, 6√5 + 2√3] / 10.

Thus the equation of that bisector is given by

r(t) = [5, 2] + t [-3√5 + 11√3, 6√5 + 2√3].

The direction of the other angular bisector is perpendicular to that of the above equation, so it can have direction vector

[ 6√5 + 2√3, 3√5 - 11√3 ],

Then its equation can be given by

r(s) = [5, 2] + s [ 6√5 + 2√3, 3√5 - 11√3 ].

b)

Yes.

Sketch two intersecting lines and then denote the two angles of intersection by α and β. Clearly, α + β = π. Pick one line; the angle of intersection of the bisectors with this line are ½α and ½β, respectively. Since ½α + ½β = ½ π, they are perpendicular.