# Calc.

Determine the value of k so that the line with parametric equations x = 2 + 3t, y = -2 + 5t, z = k is parallel to the plane with equation 4x + 3y – 3z -12 = 0

1. 👍
2. 👎
3. 👁
1. let k= a+bt
then the line is:
x = 2 + 3t
y = -2 + 5t
z = a + bt and its direction vector is (3,5,b)

If the line is parallel to the plane then it must be perpendicular to the normal of the plane, that is
(4,3,-3)∙(3,5,b) = 0
12 + 15 - 3b = 0
b = 9

so z = a + 9t

No matter what the value of a, the line will be parallel to the plane.

trying to intersect the line with the plane we get
4(2+3t) + 3(-2+5t) - 3(a+9t) = 12
the t's drop out and we get a = -10/3

so if a = -10/3 the line is parallel to but also on the plane, while for any other value of a, the line will not be on the plane, but parallel to it

so possible values of k could be just 9t, or 5+9t, or 1+9t, etc

1. 👍
2. 👎
2. The vector 4i + 3j - 3k is normal to the given plane. (k is a unit vector here, not the unknown you are looking for, which I will call K). If the line represented by the parametric equations is parallel to the plane, it must be perpendicular to the normal to the plane.

The vector parallel to the line is defined by
t = (x-2)/3 = (y+2)/5 , with no component along the z axis. The line is in the x,y plane regardless of the value of K. Its vector components are
3i + 5j +0k

One can write an equation that requires the two lines to be perpendicular, but it will not involve your unknown K, and there will be no solution, because a line with only x and y components cannot be perpendicular to the plane defined by 4x + 3y – 3z -12 = 0

Are you sure you did not omit a term that involves t in the parametric definition of the line? Is it really z = K ?

1. 👍
2. 👎
3. Reiny treats your k as an unknown a + bt parametric term, so there are really two unknowns in that case. I treat k as an unknown constant. In Reiny's czse, solutions can be obtained, as he has done.

You should have been clearer about what k is supposed to represent.

1. 👍
2. 👎
4. I just copied the question as it was...I tried asking a friend and the teacher about k. When I tried to work it out, I assumed that the direction vector was 0, because it was not k+t or any number at all, but I am still unsure. Thanks.

1. 👍
2. 👎

## Similar Questions

1. ### Pre Calculus

Identify the parametric equations that represent the same path as the following parametric equations. x(t)=2cos2t y(t)=sin3t a. x(t)=2cos2t y(t)=sin6t b. x(t)=4cos4t y(t)=sin6t c. x(t)=2cos4t y(t)=sin6t d. x(t)=4cos2t y(t)=2sin3t

2. ### vectors linear algebra

The parametric equations for a line L1 are as follows: x = −4−6t y = 2+2t z = 1−4t Let L2 be the line parallel to L1 and passing through the point (−3, −3, −3). Find the point P on L2 whose x-coordinate is −12.

3. ### Pre-Calculus

Which is the polar form of the parametric equations x=5cos(theta) and y=5sin(theta) ? a. r= 5(theta) b. r= 5 c. r= 25 cos (theta) sin (theta)***** d. r= 25cos^2 (theta) + 25sin^2 (theta) Which is the polar form of the parametric

4. ### Algebra

The parametric equations for a line L1 are as follows: x = −1−2t y = 4+4t z = 3−2t Let L2 be the line parallel to L1 and passing through the point (2, 5, −3). Find the point P on L2 whose x-coordinate is −3. P = (−3,

1. ### precalculus

For parametric equations x=a cos t and y=b sin t, describe how the values of a and b determine which conic section will be traced.

2. ### Math

Determine vector, parametric and, if possible, symmetric equations for the line through Q(2, -1, 3) and the mid-point of the line segment from L(3, -2, 5) to M(1, 4, -7).

3. ### Math

Write the parametric equations of a line perpendicular to 4x + 8y +7 =0 with the same x-intercept as [𝑥,𝑦]=[2,7]+𝑡[−10,3].

4. ### PreCAL

Write an equation in slope–intercept form of the line with the given parametric equations. x = 9t + 2 y = –6t + 9 Answer Choices (A) y=-2/3x-3/7 (B) y=-3/2x+3/31 (C) y=31/3x-2/3 (D) y=-2/3x+31/3

1. ### Math

Write an equation in slope-intercept form of the line whose parametric equations are: x = (1/2)t +2/3 and y = t - 3/4. a) y = 2x - 7/12 b) y = 2x + 7/12 c) y = -2x - 7/12 d) y = -2x - 25/12 I think it is D.....?

2. ### 12th math

A biologist determines that the path of a bee from its hive to its foraging site can be described by the parametric equations x= 4t-1 and y=2t^2+3t-4. Which of the following equations is the curve described by these parametric

3. ### math

Determine vector, parametric, and if possible, symmetric equations of the line through D(-4, 3, 6) and parallel to the z-axis. Vector form: + t Parametric: x = -4 y = 3 z = 6 + t Symmetric: x = -4 , y = 3 , z - 6 Is this right?

4. ### Mathematics

Q/ Find the parametric equations of the line joining the points P1( 1, 1, -1) and P2( 1, 2, 0)?