Which of the followings are true? 1. L:z =1+4,y = -2+ 34, z = 4- t and Lz : 1= 25, y = 3+ 8, z = -3 + 4s are skew lines. IL L, : = - and La : z + 3 = = are skew lines. :z = -6t, y = 1+ 9t, z = -3t and Ia :z =1+ 25, y = 4 – 3s, z = 8 are parallel lines. O a. I O b.I OCLI O d. l, II O e.l

Q: A, B, and C are three points on a line. AC = 8, ВС %3 4, and AB = 12. Which point must be between…

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Q: Compute ye" ds where C is the line segment from (1, 2) to (4, 7).

A: We have to compute ∫Cyexds where C is the line segment from 1,2 to 4,7. We have, x-14-1=y-27-2=t

Q: 2. Evaluate S. 3x – 6y ds for C is the line segment from (6,0) to (6,6).

A: Solution

Q: Which of the following is the distance from the point R= (1, 4, 1) to the line passing through the…

A: Option b is correct.

Q: What is the normal line to the line that passes through (-8,0) and (-1,3) 3x-7y=-24 C (x/8) +…

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Q: uate ∫c 5x^4 ds, where C is the line segment from (1,3) to(2,6).

A: In this question, the concept of line integral is applied. Line Integral The function to be…

Q: Find the distance between the following skew lines  x/2 =y−46, z=0 x-1/2 = y/2 =z+3

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Q: Which point(s) lie on the line (x, y, z) = (2, –1, 5) + n(1, 2, - 1)? a) (1,- 3, 4) %3D b) (5, 5, 2)

A: The given line is  (x,y,z)=(2,-1,5)+n(1,2,-1) compare  with general equation of line in a space

Q: 9 At what point do the following lines in R³ intersect? h: (x, y, z)=(3, 1, 2) + s(1,0,2), SER 1₂:…

A: Let's find point of intersection.

Q: Prove that given lines are skew. L1: [x, y, z] = [1, 3, 7] + t[–1, 1, 2] L2: [x, y, z] = [4, –2,…

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Q: 1.) Create a linear line from the following information: (-3,) and (4,) аnd 10

A: Calculate the slope m for the line passing through the given points.…

Q: 17. For what value(s) of k, if any, will the lines *+1_y+2 3 k 7=(6,−1,0)+t(6,1−k,2) perpendicular?…

A: Product of DRs of two perpendicular lines is 0

Q: Show that the line koining the points A(2,3, – 1) and B (3,5, — 3),C(1,2,3) аnd (3,5.7)

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Q: Find the indicated segment. 1. QB 30/4 or 7.5 2. QD 6 3. QA 10 4. QD 6 5. QD 7 6. QD 11/2 or 5.5 7.…

A: I am attaching image so that you understand each and every step.

Q: Compute yds ds where C is the line segment from (1, 2) to (4, 7).

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Q: Show that these two lines are coincident. L1: [x, y, z ] = [1, 5, − 2] + s [1, 7, − 3] L2: [x, y, z…

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Q: 6. Use the Side-Splitter Theorem to find z given that NQ is parallel to OP. 2.5 5 M O 4 3

A: We have to find out

Q: The commentary of this proof contains two mistakes: WAR 2. M → R 3. W + M .. W v R 4. W VE 1 5. M HE…

A: Consider the given proof sequence. The premises are as follows. 1. W∧R2. M→R3. W↔M The conclusion is…

Q: If ?1 is the line through ?(1,2,7) and ?(−2,3, −4) and ?2 is the line through ?(2, −1,4) and ?(5,7,…

A: Given that 1 is the line through (1, 2, 7) and (−2, 3, −4). So the direction ratio of line 1 is =…

Q: 9. Determine the point of intersection of the following two lines: L₁: (x, y, z) = (1, 2, 1) + t(3,…

A: Find value of parametric by equating L1  and L2.

Q: 5. Determine the angle 0 between the lines 3x + 4y - 6 =0 and 2x + 3y + 4 = 0.

A: Here we have to determine the angle θ between the lines 3x+4y-6=0 ..................... (1) and…

Q: Which of the folowing is a true statement about skew lines? A. Skew lines intersect at a single…

A: It is known that skew lines are the lines that do not intersect each other and they are  in…

Q: c ls tE line cregmentfoom (3,4,0) to | (1,4,2) computef (zty?)ds&dp

A: The curve C is the line segment from 3,4,0 to 1,4,2. The parametric equation of this curve is given…

Q: What is the normal line to the line that (passes through (8, 0) and (-1,3 3x - 7y = - 24 O (x/8) +…

A: We will first find the slope of the line that passes through the two given points. The slope of the…

Q: Find the best-fit line associated with the set of points. (1, 5) (3, 14), (5, 24), (7, 0) O y = 2.0x…

A: Consider that x is the independent variable and y is the dependent variable. The size of the…

Q: O -L -5) O4-

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Q: Which of the following points is closest to (1,0, 1,0, 1, 1) using Hamming distance? O (1,1,1,1,1,0)…

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Q: Q1 find acubie palonemiat ofk given-points for27,3},(2,12)e

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Q: What is the normal line to the line that passes through (8 , 0) and (-1, 3) 3x - 7y = - 24 O (x/8) +…

A: we have to find out the equation of normal line  given points are a(8,0)  b(-1,3) of simple equation

Q: Suppose a line runs through two points: p=(3,2) and Q=(15,8) What the slop of the line

A: Given: A line passes through two points  P = 3, 2Q = 15, 8 To find: The slope of the line.

Q: Consider a crowded beach where the population of beach-goers is uniformly distributed across its…

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Q: Express the following as a linear combination of u = (4, 1, 6), v = (1, -1, 6) and w = (1, 6, 7).…

A: The objective is to find (7, 10,7) as linear combination of u, v, w.

Q: The equations [2, 1, 6] + t[6, 1, 4] and [4, 1, 3] + s[2, 10, 9] Are parallel Intersect at one point…

A: Topic = Lines Given lines are skew lines.

Q: The figure in he d agram is composed of two lines, a ray, and a segment. 139.8 40.2 3. 木 56 19.3 9…

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Q: 10. Show that these two lines are coincident. [x, y, z]= [1, 5, -2]+s[1, 7, -3] [x, y, z]=[-3, -23,…

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Q: Question 5. , Which of the following statements is true about the lines L1 : x= 4 – 2t, y = 2+ 2t, z…

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Q: Find the distance between the skew lines i(t) = (3, 0, – 2)t + (7, 9, 0) and p(s) = (4, – 4, – 5)s +…

A: We have to find the distance between skew lines.

Q: Express w as a linear combination of u and v: ? = (4,1), ? = (−2, −1), ? = (2, −2)

A: Let w = (4,1) u = (-2,-1) v=(2,-2) we can write this as: u-21 + v2-2 = 41   -2u + 2v = 4 u - 2v = 1…

Q: Consider the points a, b, c, d and the lines L, M in the following figure.

A: Given a graph containing the points a, b, c, d and lines L, M

Q: Find the distance between the skew lines r(t)=⟨3,−2,−4⟩t+⟨−4,−2,5⟩ and p(s)=⟨5,2,1⟩s+⟨−6,4,7⟩

A: Given r(t)=⟨3,−2,−4⟩t+⟨−4,−2,5⟩ and p(s)=⟨5,2,1⟩s+⟨−6,4,7⟩

Q: Consider the points P=(20,n,21) and Q=(10,0,10). n < 0 y ||PQ|| = V230 %3D If it holds that then it…

A: The points P=20, n, 21 and Q=10, 0, 10Where, n&lt;0If PQ→=230Then to find the true statement about…

Q: Compute yds where C is the line segment from (1, 2) to (4, 7).

A: We know that,       . Now,

Q: 3. Let A= (1,2,4), B=(4,1,5), C=(4,2, 6), D=(7,1,7) Find the exact distance between the lines and…

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Q: Explain why the lines below are skew. l1: (x, y, z) = (-2, – 2, – 2) + n(1, 2, 3) and l2: (x, y, z)…

A: To show that the given two lines are skew lines. The two lines are                          l1 = (-2…

Q: What is the point of intersection, P(x, y) of the following lines: 5x + 2y = 4 and -2x + y = 11. x =…

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Q: 13- Find the eqation of the line that passes through A(1,2,1) and intersects the line 끌=D글3D a righ…

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Q: Let l: = = -: - 5 and p = (1,7, -4) + A(1, -2, 2) be two lines in space.

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Q: Show the lines (x, y, z) = (5 + 2t, 3 + 2t, 1 – t) and (x, y, z) = (13 – 3r, 13 + 2r, 4 + 4r) are…

A: Given:- Show the lines (x, y, z) = (5 + 2t, 3+2t, 1- t) and (x, y, z)= (13-3r, 13 + 2r, 4 + 4r) are…

Q: (d) The lines 1(1) = (1 – t, 1 + 1, 21) and I2(t) = (4 – 21,3 + 21, 1 + 5t) are parallel. True False

A: Let                          L1(t) : (a1-b1t, c1-d1t, e1-f1t)       L2(t) : (a2-b2t, c2-d2t, e2-f2t)…

Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.