(a) Let ri(t) = [2, 1, – 1] + t[1, 1, 1] be the equation of a line L1 in R³. De- termine the equation of a line L2 that is parallel to L1 and contains the point A(1, 2, 3). (b) Find the vector equation for the line L3 through B(1, –3, 1) that is par- allel to both of the planes P1 : x + y + 2z = 1 and P2 : x – y + 3z = 2.

Q: = (4, 1, a), Q = (–2, 0, b) and v = i – 3j + 4k be two points and a vector in R³ respectively. (a)…

A:

Q: Let T: R2 R2 be a reflection in the line y = x. Find the image of each vector. (a) (3, 1)

A: (a)  (1, 3) (b)  (-1, 0) (c)  (0, -a) (d)  (b, 0) (e)  (-d, c) (f)  (g, -f)

Q: Compute the orthogonal projection of v= 7 onto the line through 4 2 and the origin.

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Q: 4. Write the vector equation for a line parallel to the line L: F = (1,-7)+t(2, -3), t € R…

A: Introduction: In three-dimensional geometry, a line can also be described through a vector equation,…

Q: 1. Consider the pointsS= (1,−3) andT= (2,5).(a) Sketch the vector−→ST.(b) Give the component form…

A: Here 2 different problems are provided. As per our Bartleby Guidelines, we are supposed to…

Q: Compute the orthogonal projection of v = proj, (v) = -5 onto the line L through -4 and the origin. 6…

A: Introduction:  The vector projection of a vector a on (or onto) a nonzero vector b, sometimes…

Q: 1. Find a unit vector perpendicular to the line passing through the points (1, 0, -3) and (2, 1,…

A:

Q: A vector equation for the line through the points A (3, -4, -3) and B (3,4,2) is (x, y, z) = . (You…

A: Concept to be used:If a line passes through two points A,Bwith position vectors aand brespectively,…

Q: (b) (c) Given that a(x)=0, show that ã•(¯ × (c − ã))=ã ● (ã×b). Hence, given that à =, b = and c =,…

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Q: Four students each made a statement about the equation of a line in R² given in vector form, 7 =…

A: The equation of a line on the Cartesian coordinate system can be represented by the linear equation:…

Q: kpression for "12 times x."

A: Times means multiplication. Sometimes it is denoted by dot.

Q: A line, in R³ which passes through the point (xo, yo, zo) and with the direction vector (a, b, c),…

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Q: Find the vector equation of the plane that contains the parallel lines [x, y, z]= [0,1,3] + [-6,…

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Q: 22. Given the symmetric equations t = - and t = +', find the vector equation of the line.

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Q: Let T: R2 R be a reflection in the line y = x. Find the image of each vector. (a) (0, 8) (b) (1,-1)…

A: Let T:R2→R2 be a reflection in the line y=x.

Q: Find an equation of the plane that contains the point (-1, 2, 3) and is perpendicular to the line…

A: The plane is perpendicular to the line, then the direction vector of the line is normal to the…

Q: Let Lz be the line passing through the points Q1=(1, 3, 1) and Q2=(7, 12, 4). Find a value of k so…

A: We will solve the problem

Q: Exercise 1. A student is asked to prove If m is even and n is odd, then m² + n² is odd. 2 The…

A: Let us answer above questions

Q: The point A has position vector i+5k and the line l1 has vector equation r=7i+4j–2k+ a(5i+3j+k). (a)…

A: The position vector of point A is given by i+5k. The line l1 has vector equation…

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.