Mathematics, 21.02.2020 00:03 ayoismeisalex

Find an equation of the plane.

The plane through the point (2,0,1) and perpendicular to the line x = 3t, y = 2-t, z = 3+4t

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The equation of the plane which contains the point (2, 0, 1) and perpendicular to the line $(x,y, z) = (0, 2, 3) + t\cdot (3, -1, 4)$ is $3\cdot x - y +4\cdot z = 10$ .

First, we use the dot product between the vector slope and a vector that pass through and parallel to the plane to derive the equation of the plane:

We determine the vector slope associated with the line:

$(x,y, z) = (0, 2, 3) + t\cdot (3, -1, 4)$ (1)

By direct inspection, we see that vector slope is (3, -1, 4) .

Then, we construct the following vector equation by means of the dot product between the vector slope and a vector passing through and parallel to the plane:

$(x-2, y, z-1)\,\bullet\,(3, -1, 4) = 0$

$3\cdot (x-2) -y +4\cdot (z-1) = 0$

$3\cdot x -6 -y +4\cdot z -4 = 0$

$3\cdot x - y +4\cdot z = 10$ (2)

The equation of the plane which contains the point (2, 0, 1) and perpendicular to the line $(x,y, z) = (0, 2, 3) + t\cdot (3, -1, 4)$ is $3\cdot x - y +4\cdot z = 10$ .

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Answer from: michellemitchel

Consider the vector equation for the given line,

$\vec r(t)=\langle3t,2-t,3+4t\rangle$

which has tangent (direction) vector

$\dfrac{\mathrm d\vec r(t)}{\mathrm dt}=\langle3,-1,4\rangle$

The normal vector to the plane is parallel to this tangent vector, so we can use it as the normal vector itself.

Then the plane has equation

$\langle3,-1,4\rangle\cdot\langle x-2,y,z-1\rangle=0$

$\implies3(x-2)-y+4(z-1)=0$

$\implies\boxed{3x-y+4z=10}$

Answer from: Quest

b. 6

step-by-step explanation:

we are given the table representing an exponential function.

we can see from the table that, as x is increasing, the value of y is increasing in the power of 6.

i.e. the following relation is obtained,

x = 1, y = 6

x = 2, y = 36 = $6^{2}$

x = 3, y = 216 = $6^{3}$

x = 4, y = 1296 = $6^{4}$

x = 5, y = 7776 = $6^{5}$

so, the geometric series obtained is $6, 6^{2}, 6^{3}, 6^{4}, 6^{5}$ .

thus, the common ratio is given by $\frac{6^{2}}{6}$ = $\frac{36}{6}$ = 6.

hence, the common ratio is 6.

Answer from: Quest

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240

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Find an equation of the plane.

The plane through the point (2,0,1) and perpendicular to...

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