Question

The equations of three lines are given below Line 1: y=3x-5 Line 2: y=-1/3x +1 Line 3: 3x-9y=18 For each pair of lines, determine whether they are parallel, perpendicular, or neither: 1. Line 1 and 2 are they (parallel, perpendicular, or neither.) 2. Line 1 and 3 are they (parallel, perpendicular, or neither.) 3. Line 2 and 3 are they (parallel, perpendicular, or neither.

Answer #1

Solution :

Line 1: y=3x-5

Line2 : y=-x/3+1

Line3 : 3x-9y=18==>y=x/3-2

Now comparing with equation

y=mx+c , where m is slope of Line, c is y-intercept

So

Slope of Line 1 say m_1=3

Slope of Line 2 say m_2 =-1/3

Slope of Line 3 say m_3 =1/3

Result 1) Two lines are parallel if they have same slopes

Result 2) Two lines are y=mx+c and y=m'x+c are perpendicular if mm'=-1

now m_1.m_2=-1===>line L_1 and L_2 are perpendicular

Line L_1 and L_3 are Neither parallel nor perpendicular {as their slopes are not equal and product of slopes is not equal to -1}

Lines L_2 and L_3 are Neither parallel nor perpendicular {as their slopes are not equal and product of slopes is not equal to -1}

1. Determine whether the lines are parallel, perpendicular or
neither. (x-1)/2 = (y+2)/5 = (z-3)/4 and (x-2)/4 = (y-1)/3 =
(z-2)/6
2. A) Find the line intersection of vector planes given by the
equations -2x+3y-z+4=0 and 3x-2y+z=-2
B) Given U = <2, -3, 4> and V= <-1, 3, -2> Find a. U
. V b. U x V

Examine whether or not these pair of lines are perpendicular to
each other. (1) y - 3x - 2 = 0 and 3y + x + 9 = 0 (2) 3y - 4 = 2x +
3 and y-5 = x+ 6 (3) Find the equations of the tangent and normal
to the curve xsquare + ysquare+3xy-11 = 0 at the point x = 1, y =
2.

(a) Find the distance between the skew lines l1 and l2 given
with the vector equations l1 : r1(t) = (1+t)i+ (1+6t)j+ (2t)k; l2 :
r2(s) = (1+2s)i+ (5+15s)j+ (−2+6s)k.
(b) Determine if the plane given by the Cartesian equation −x +
2z = 0 and the line given by the parametric equations x = 5 + 8t, y
= 2 − t, z = 10 + 4t are orthogonal, parallel, or neither.

Write equations of the lines through the given point parallel to
and perpendicular to the given line. 6x − 2y = 7, (6, 1)

Show that the two lines with equations (x, y, z) = (-1, 3,
-4) + t(1, -1, 2) and (x, y, z) = (5, -3, 2) + s(-2, 2,
2) are perpendicular. Determine how the two lines
interact.
Find the point of intersection of the line (x, y, z) = (1,
-2, 1) + t(4, -3, -2) and the plane x – 2y + 3z =
-8.

1. Solve all three:
a. Determine whether the plane 2x + y + 3z – 6 = 0 passes
through the points (3,6,-2) and (-1,5,-1)
b. Find the equation of the plane that passes through the points
(2,2,1) and (-1,1,-1) and is perpendicular to the plane 2x - 3y + z
= 3.
c. Determine whether the planes are parallel, orthogonal, or
neither. If they are neither parallel nor orthogonal, find the
angle of intersection:
3x + y - 4z...

Find sets of parametric equations and symmetric equations of the
line that passes through the given point and is parallel to the
given vector or line. (For each line, write the direction numbers
as integers.) Point Parallel to (−6, 0, 2) v = 3i + 4j − 7k
(a) parametric equations (Enter your answers as a
comma-separated list.)
(b) symmetric equations 3x = y 4 = 7z x 3 = y = z 7 x + 6 3 = y...

Find parametric equations of the line that passes through (1, 2,
3) and is parallel to the plane
2 x − y + z = 3 and is perpendicular to the line with parametric
equations:
x=5-t, y=3+2t , z=4-t

Find parametric equations of the line that passes through (1, 2,
3) and is parallel to the plane
2 x − y + z = 3 and is perpendicular to the line with parametric
equations:
x=5-t, y=3+2t, z=4-t

Find equations of the tangent lines to the curve y = x − 1/x + 1
that are parallel to the line x − 2y = 3. y= y=

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