**Finding the Plane Equation****Problem:**Determine the equation of a plane that passes through the point (6, 7, -2) and contains the line given by the parametric equations:\[ x = 4 - 2t, \ y = 3 + 8t, \ z = 7 + 4t \]**Solution:**1. **Given Data:** - Point: $(6, 7, -2)$ - Line: $x = 4 - 2t, \ y = 3 + 8t, \ z = 7 + 4t$2. **Direction Vector of the Line:** The parametric equations can be written as: \[ \vec{r} = \langle 4, 3, 7 \rangle + t \langle -2, 8, 4 \rangle \] The direction vector is: \[ \vec{d} = \langle -2, 8, 4 \rangle \]3. **Vector from Point on the Line to the Given Point:** Let $ P = (6, 7, -2) $ be the given point and $ Q = (4, 3, 7) $ be a point on the line when $ t = 0 $. Vector $ \vec{PQ} = \langle 6 - 4, 7 - 3, -2 - 7 \rangle = \langle 2, 4, -9 \rangle $4. **Normal Vector to the Plane (Cross Product):** Compute the normal vector $ \vec{n} $ to the plane which is perpendicular to both $ \vec{PQ} $ and the direction vector $ \vec{d} $: \[ \vec{n} = \vec{PQ} \times \vec{d} \] Calculate the cross product: \[ \vec{n} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & 4 & -9 \\ -2 & 8 & 4 \\ \end{vmatrix} \] \

Recall: General equation of a plane : An equation for the plane through (x0, y0,z0 ) and…

Answered: 27 hPlans that pas Ses through the…

Math

Calculus

27 hPlans that pas Ses through the Point C60,27 and contains the like メt4-2t)ソ= 34st) Zzス+4t 7.

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

See similar textbooks

Related questions

Q: What is the latus rectum of the equation 9x^2-y^2-2y-10=0?

Q: limx→∞ 4x² 5x - 8 - 2x

Q: x²+6x+9 x+1 b) x = -6 4. Find the zeroes of f(x) = a) x = -3 c) x = -2 d) x = -5

A: This is a problem related to calculating roots of the equation.

Q: PX)= 9 (x+3)" (メー1)2 and aco Find the end behavior

Q: Ir te line x34+2¢, もー4t paralled he plane 3x +y+そこ子-

Q: 4. A line segment L passing through the point (1, 1) has its endpoints lying on the x-axis and…

Q: Solve using undetermined coefficients: (D^4 + 4D^2)y = 48x + 96sin2x

Q: if one root is 2, find each value of k and the other roots A) 25x^4+kx^2+16=0 B) 3x^3-15x^2+kx-4=0

Q: 17x^(2)+7y^(2)=119. Find the length of a, b, c.

Q: Find and classifythe extreme Point of the function f(x,y)= 3yx+x -3*2

A: Given function, fx,y=3y2x+x3-3y3-3x2+2 Partially differentiating w.r.t. 'x' to both sides-…

Q: What is the value of positive number of C if a, and b are also any positive numbers in the equation:…

A: The equation is defined as the mathematical expression which is used for two statements connected…

Q: Solve the equation. Make sure to check for extraneous solutions A) 8^m-7=92 B) 10 (5^3k-3)=40 C)…

Q: write the given expression in the form of a^2 - u2: 15 + 2t - t^2 is equal to. SUBMIT YOUR SOLUTION…

A: Given: 15+2t-t2 To write in the form: a2-u2 15+2t-t2 = -t2-2t-15

Q: Solve using quadratic formula: z^2 + 2z+ 2*

Q: find the midpoint for A( 2 , 8 ) B( -2 , 2 )

A: We can midpoint formula of straight line.

Q: Find the missing endpoint. Endpoint: (2,5), midpoint: (5,1)

A: Given, Endpoint is 2,5 and midpoint is 5,1.We know that, The midpoint is…

Q: The center of the geometric figure described by 16x^2-160x+36y+292=0 is at what point?

A: We have the given function as 16x2-160x+36y+292=0 On solving further by using completing the square…

Q: ), (4,2), (5,3) b

A: Given as, Here,

Q: find the midpoint for A( -5 , 0 ) B( -15 , 6 )

Q: As for the relationship: y= 4x^4 - 2x^2, it is symmetrical to: a.y-axis b.x-axis c.origin d.none

A: The given function y=4x4-2x2 This is an even function because it satisfies the expression: fx=f-x…

Q: find the fur pose of the function- and give the reason. メ * x²4+ 3× メ<t

A: Follow the procedure given below.

Q: (2x+7 14 い-3 f -3くろくて 1-22+414 xフて 5.- Sketcch the graph of ta)=

Q: Find the midpoint between (6,7) and (6,-5)

Q: what quadrtant is the point (-3,2)

A: Point (-3,2) Here x axis is negative and y axis is positive

Q: The equation ℎ = −2t^2 + 16t + 9 represents the height of a ball where h, is the height of the ball…

A: (a) h(t) = - 2t² + 16t + 9 So, h'(t) = - 4t + 16 So, h"(t) = - 4 Now, for extreme values of h, we…

Q: b) y = 2x √x+1

Q: value

Q: f(x)%=- 2+8x-8

Question

**Finding the Plane Equation**

**Problem:**
Determine the equation of a plane that passes through the point (6, 7, -2) and contains the line given by the parametric equations:
\[ x = 4 - 2t, \ y = 3 + 8t, \ z = 7 + 4t \]

**Solution:**

1. **Given Data:**

- Point: $(6, 7, -2)$
- Line: $x = 4 - 2t, \ y = 3 + 8t, \ z = 7 + 4t$

2. **Direction Vector of the Line:**

The parametric equations can be written as:
\[
\vec{r} = \langle 4, 3, 7 \rangle + t \langle -2, 8, 4 \rangle
\]

The direction vector is:
\[
\vec{d} = \langle -2, 8, 4 \rangle
\]

3. **Vector from Point on the Line to the Given Point:**

Let $ P = (6, 7, -2) $ be the given point and $ Q = (4, 3, 7) $ be a point on the line when $ t = 0 $.

Vector $ \vec{PQ} = \langle 6 - 4, 7 - 3, -2 - 7 \rangle = \langle 2, 4, -9 \rangle $

4. **Normal Vector to the Plane (Cross Product):**

Compute the normal vector $ \vec{n} $ to the plane which is perpendicular to both $ \vec{PQ} $ and the direction vector $ \vec{d} $:

\[
\vec{n} = \vec{PQ} \times \vec{d}
\]

Calculate the cross product:
\[
\vec{n} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
2 & 4 & -9 \\
-2 & 8 & 4 \\
\end{vmatrix}
\]

\

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Step by step

Solved in 4 steps with 3 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Dilations$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

Similar questions

Eliminate the arbitrary constant in the equation: a.) x²y = 4+Cy b.) y-x = C1e^x + C2e^-2x c.) y = 2C1cos(2theta) + C2sin(2theta) show complete solution
Show by completing a square that the solution to the equation ax^2-bx-a=b are x=-1 and x=(a+b)/a
What is the midpoint between the endpoints (5,9) and (-3,-7)
What are coordinates of AABC with A(-4,2), B(4,4), and C(0,-2) after a dilation by a factor of 0.5?
and てつす (x²9) 2xCx2+27). find all asympoles. aire 3. {मा) नणणप्जग Complele first dorivalive (inteuale of ncrease and second derivatie anaulsis Cintervals of concauity and inflectiom points) and skelch a 2pm function.
Show the complete solution, do not shortcut please!
Solve Q4.
Find the shape using the discriminate form (B^2 - 4AC) of the equation y^2 - 6y - 4x^2 - 8x = 95.
Find x-intercept and y-intercept with solution.