Express (-1,2,4) as linear combination of x = (-1,2,0), y = (0,-1,1) and z = (3,-4,2) in the vector V3 of real numbers. %3D

Q: Find the dot product of the pair of vectors (-2, 5) and (4, 1).

A: Given : < -2 , 5 > and < 4 , 1 >

Q: Write the vector (-5, -9) as a linear combination of the vectors (-2, 3) and (1, 4).

A: NOTE: Refresh your page if you can't see any equations. . write both vectors in the matrix and the…

Q: Express the vector (-1,5,-6) in R'as (2,0,7), (2,4,5) and (2,-12,13). linear combination of

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Q: Consider the vectors (2, 1, 0, 2], [3, 1,0, 1], [1, 1, 0,-1] in R'. Decide whether they independent.…

A: Given: Vectors in ℝ4,                              2,1,0,2, 3,1,0,1, 1,1,0,-1 To find: Whether the…

Q: Express the vector (-1, 0) as a linear combination of the vectors (-1, - 1) and (-1, 1). (-1,0) =…

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Q: Determine whether the set of vectors is linearly independent: {(1,0,1), (3,4,-3), (2,2,-1)}.

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Q: Find the sum of the given vectors. a - (3, 0, 2), b- (0, 6, 0) Illustrate geometrically.

A: Given, Two vectors  a = ⟨3, 0, 2⟩   b = ⟨0, 6, 0⟩  We have to find the sum of vectors and illustrate…

Q: Graph the two vectors x= (5, 1,3) and y=(-1,3,1

A: Topic:- 3D

Q: Explain “ some combination gives the zero vector, other than the trivial combination with every…

A: The statement is “ some combination gives the zero vector, other than the trivial combination with…

Q: In each case, find the linear combination of the first two vectors that is as close as possible to…

A: Both part of the question can be converted into the question based on least squares solution of a…

Q: Express the vector (-1,5, –6) in R³as a linear combination of (2,0,7), (2,4,5) and (2, –12,13).

A: Solved below

Q: Reflect the vector (-3, 5) across the line y=x

A: The vector is (-3,5)

Q: Find a vector equation for the line through the point (-3, –1, 8) perpendicular to the vectors ū =…

A: Vector equation of the line passing through point (a,b,c) and perpendicular to vectors u→ and v→ is…

Q: Find two vectors that are not multiples of each other and are both orthogonal to〈2, 0, −3〉

A: Consider the first vector, and the second vector, x1,y1,z1 and x2,y2,z2, respectively. Since both…

Q: Find axb, a vector orthogonal to both vectors a' and b. a = (-3, 2, 1) and =(4, -2,5)

A:

Q: Determine if the vectors , and are linearly independent.

A:

Q: Find an equation of the plane that passes through the point P(-4, -5, 4) and has the vector n =…

A: Plane passes through the point P-4, -5, 4 and has the vector n=-4, 9, 4 as a normal.

Q: Every vector of R²is linear combination of (-1, 1), (2, –2)? Justify

A:

Q: Identify if the vectors〈-1,2,-1〉and 〈2,-4,-1〉parallel or not

A: Two vectors are parallel if the ratio of each component are equal.

Q: Write the vector (7,8, –8) as a linear combination (if possible) as of the vectors (-1,1,2), (2,3,…

A: See answer below

Q: Find the vector v = PQ. Р- (2, 1,4) Q = (-8.25, -4.25, 8) v =

A:

Q: Find the sum of the given vectors. а - (5, -1), ь - (-1, 8) a +b = 4,7 Illustrate geometrically. y

A: We have

Q: If the components of a contravariant vector in (x') coordinate system are (3,4). Find its components…

A: Given: Components of contravariant vector in xi coordinate system is (3,4)=x1,x2 Coordinate for…

Q: Determine the tail of the vector whose head is (3, −2, 2).

A: We have to determine the tail of the vector whose head is 3, -2, 2 and position vector is: 24-1 We…

Q: Find a direction vector for the line containing the points (-1, 5, 2) and the point (2, 3, - 1).

A: We have to find a direction vector for the line containing the points -1, 5, 2 and 2, 3, -1. We know…

Q: Find a vector equation of a line passing through the point (-1, -2, 4) and perpendicular to both a =…

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Q: Express the vector x as a linear combination of the u's. U₁ = 0, U₂ = 5, -1.4 U3 Ox=-3u₁ +4u2 +243…

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Q: Project the vector b = (1,3,1)" onto the line through %3D a13D (-1,2, 2)Т, (2, 2, –1)", аz 3D az=…

A: Given: Let us assume, b= (1,3,1)T  a1=(-1,2,2)T   a2=(2,2,-1)T   a3=(2,-1,2)T

Q: Express the vector v =-1 as a linear combination of the vectors -3 0,v2 4 and v3 -4 , if possible.…

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Q: Write the vector (-5, -9) as a linear combination of the vectors (-2, 3) and (1

A:

Q: Find a vector equation for the line that goes through the points (7, 6, 2) and (2, 14, 2). F(t) =…

A: Given: Let A(7,6,2) and B(2,14,2) be the two points the line passes through To find: vector equation…

Q: Use vectors to determine whether the points (3,4,-1), (-1,6,9) and (5,3,4) are collinear.

A: The points (3,4,-1), (-1,6,9) and (5,3,4)

Q: 3. If a = 3x - 5y and b=-2x+9y, write the vector 5a-8b in terms of x and y

A:

Q: Write down the vector equation of the line that passes through the points (x,y,z) = (2,3,7) and…

A: The given problem is to find the vector equation of line that passes through the given two points.

Q: Find symmetric equations for the line that passes through the point (5, -2, 7) and is parallel to…

A:

Q: Write the component form of a vector v with initial point P(8.6, -2.5) and terminal point Q (6.7,…

A:

Q: Write the vector (7, 8, -8) as a linear combination (if possible) as of the vectors (-1, 1, 2), (2,…

A: Let the linear combination be, x-112+y23-2=78-8 It is required to determine the value of 'x' and 'y'…

Q: Find an equation of the line that goes through A(0, 1, 4) and is orthogonal to the Y-axis and vector…

A:

Q: Find the indicated vector. Let u - (-3,-7) --8, -1). Find -2u - 3v.

A: To find -2u-3v :

Q: What is the component form of the vector represented by 2a when = (-2, 3)? Enter the answer in the…

A: If we multiply by a constant to any vector then  magnitude of that vector will be changed.

Q: Find two vectors ūj and īz whose sum is (-5, –3), where vj is parallel to (2, 5) while īz is…

A: To find - Find two vectors v1¯ and v2¯ whose sum is -5, -3, where v1¯ is parallel to 2, 5 while v2¯…

Q: Find two vectors V1 and V2 whose sum is ⟨2,−5,3⟩, where V1 is parallel to ⟨4,4,0⟩ while V2 is…

A: Parallel vector will be multiple of each other  Perpendicular vector has dot products zero

Q: Write the vector (7, 4,-12) vectors (-2, 1, 2), (1, 1, –2) as a linear combination (if possible) as…

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Q: Show whether the vector w is a linear combination of vectors v1, v2 and v3 where --I --E --4 -- w =…

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Q: .3

A: Given: v={ √2,3√2}

Q: Find a vector equation for the line segment from (2, -1, 4) to (4, 6, 1)

A:

Q: Find the component form of the vector.

A: Consider the figure: Find the component of form of a vector v.   The component form of a vector in…

Q: Determine the vector equation for the line that is parallel to = [1,5,-7]+[2,-1,5] with the same…

A: The general equation of line in vector form is    .Then note the result : Line passing through the…

Q: Solve for the linear equation of the plane through the origin and perpendicular to the vector…

A: Topic:- geometry

Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.  

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.