2 A crystal has a basis of one atom per lattice point and a set of primitive trans- lations vectors (in Aº): a = 3i, T = 3), , T = 1.5(i + 3 + Â), where i, j and k are unit vectors in the x, y and z directions of a Cartesian coordinate

Q: Find a unit norm vector that is orthogonal to both the vector (1,2, –1) and the line of intersection…

A:

Q: 7. Find an initial point P of a nonzero vector u = PQ with ter- minal point Q(3, 0, –5) and such…

A: We have to find an initial point P of a non zero vector u = PQ→,  with terminal point Q(3, 0, -5)…

Q: Find a unit vector normal to the plane containing v=2i+3j-k and . w = -2i - 4j - 3k.

A: Given: v=2i+3j-k  w = -2i - 4j - 3k.

Q: 2. Choose the linearly dependent vectors in R

A: two non zero vectors V1, V2 are said to be linearly independent if and only if aV1+bV2= 0 ⇒ a=0 and…

Q: Consider tne vectors A vector W =(x,x,y) in R that simultaneously satisfies the following…

A:

Q: 5. Show that the vectors a = 3i- j+4k, b = i- 3j – 2k and c = 4i – 3j+2k are linearly independent.…

A:

Q: (2.3) Prove that each of the following systems a) Parabolic cylindrical coordinates (u, v, z)…

A: The given question can be solved as shown in step2.

Q: v, =,v, =, v, =

A: To add or subtract two vectors, add or subtract the corresponding components of each vector. For…

Q: In the space referred to a direct orthonormal system (0,i, j, k), given the points: А (1,1, 2), В…

A: 4) Volume of tetrahedron is 16a→ b→ c→ where a→,b→ and c→ are position vectors any three points with…

Q: For what values of the scalar t are the three row vectors (t,3,0), (3,t,3) and (0,3,t) linearly…

A:

Q: 1. Given the vectors vi = (1,2, 1), v2 = (1, 1, 1) and v3 = (1,0,0). Transform the vectors into…

A:

Q: Find volume V(S) of the parallelepiped S in R3 determined by the (3,4,1); v3 following vectors: vị =…

A:

Q: Obtain a vector orthonormal to the vectors i−j  and (1, 0, 1)

A:

Q: i. Start with finding Two vectors u and v of R' which are linearly independent (you can see this by…

A: In the given question we have to find the the Linearly independent vectors in ℝ3. Concept: In a set…

Q: Find a basis for the plane 3r - y+1z 0. withe the vectors required to be orthogonal to each other.…

A: Let's find.

Q: 2. Consider the vectors in En : e, = (1; 0; 0); ez = (0; 1; 0); ez = (1; 0; 0); Write down the…

A: Solution :-  the given vecots are e1 = ( 1; 0; 0 )  , e2 =  ( 0; 1; 0 )  , e3 = ( 0; 0; 1 )  and  (…

Q: ors a = 6i + 7j + 6k and b = 4i − j + 8k, where i, j, and k are mutuallyperpendicular unit vectors…

A: In this question, the concept of the dot product is applied. Dot Product The product of the…

Q: In the space referred to a direct orthonormal system (0,i, j, k), given the points: A (1,1, 2), B…

A: We use vector and Three Dimensional Geometry formula to solve these questions.

Q: 2. Let u e R" and v E R" be two non-zero vectors, in other words at least one component of the…

A:

Q: 1. Find a basis for the orthogonal Conplement of the Subs pace of IR? Spanned by the vectors…

A:

Q: 24. If A and B are nonzero vectors, prove that the vector A- cB is orthogonal to B if e= A· B/|B.…

A:

Q: 1. Show that the following vectors are linearly dependent. Uz = U3= %3D -2 3.

A:

Q: Find the scalar and vector projections of b onto a, where a=⟨−1,1,2⟩ and b=⟨−1,5,4⟩

A:

Q: Find a basis for the set of vectors in ℝ3 in the plane × + 2y + z = 0. [Hint: Think of the equation…

A:

Q: 1 Find the vector projection of u onto v. u= 3i- 5j ++ 2k v=7i+j- 2k

A: Please refer the attached image for complete solution.

Q: 5. (a) Consider the vectors u = (1,0, 1) and ē of the 2-dimensional parallelogram determined by…

A: we are given that

Q: 3 Find the coordinates of y 7 relative to the orthogonal basis Uj = U2 = U3 = -2 -4 10 2 2 4 252

A: The solution is given as

Q: 4 Suppose v1, V2, V3 are vectors in R*, and the matrix A = v, v2 V3 has rank 1 W = Span{v1, V2, V3}…

A:

Q: Find x such that the vectors xi+2j -3k, i+j ,j+3k are Coplanars (lies the Same Plane) . on

A:

Q: 5. A system of forces F,, E, and F, acting through points with position vectors r,r, and r,…

A: (i)…

Q: 1 Find two vectors of norm 1 that are orthogonal to the vectors u = and v = 3

A: Given problem is :

Q: 1. Determine if the vectors e2t and e-t are linearly inde- e2t e-t pendent/dependent.

A:

Q: Given the 3-dimensional vectors: A = (xy)i + (2yz)j +(3xz)k B = (yz)i + (2xz)j +(3xy)k Determine the…

A: Here given vectors as A=xyi+2yzj+3xzkB=yzi+2xzj+3xyk

Q: what are the components of a vector? what if the initial point is not (0,0)? given a vector v and a…

A: A scalar is a real number. A scaler is a single, independent number.  A vector is a ordered…

Q: 26. Find a unit vector whose representations are perpendicular to the plane containing PQ and PR if…

A:

Q: Let x = (1,2,3) and y = (1,1,1) Project y on x (in econometrics, this may be denoted by Px). Are the…

A: We will first write the formula for projection of a vector onto another vector b .

Q: Find a nonzero vector u with initial point P(-1, 3, -5) such that u has the same direction as v =…

A:

Q: 5. Show that the vectors a = 3i-j+ 4k, b =i-3j–2k and e = 4i– 3j+2k are linearly independent. Find…

A:

Q: Find a basis for the set of vectors in R^3 in the plane 3x - 2y + z = 0. Also state the dimension.

A:

Q: Find the scalar equation for the plane passing through the points P1(2, 1, 5), P2(2, 0, 9), and…

A:

Q: 6.) Find a vector equivalent to the vector PQ with its initial point at the origin. = (;-2), Q =…

A:

Q: b/Let u and v be a nonzero vectors in 3- space, and let e be the angle between these vectors when…

A: Let u and v be tthe vectors in 3-spaceThen by Lagrange's identity,we haveu×v2 =u2.v2-(u.v)2…

Q: 4. Determine whether the vectors a = 2i – 3j+ 5k, b = 3i+j – 2k and c =i- 7j +12k linearly dependent…

A: To check whether the vectors are linearly dependent or linearly independent.

Q: 2 3. (a) Find two vectors of norm 1 that are orthogonal to the vectors u = -3 and v = .3 4

A: What is the inner product of the vector space of 2 by 2 matrices. We assume it is Frobbenius inner…

Q: vectors form a basis of the R³ vector space? (-)-() 7 4 4 5 ctors y₁, y₂, Y3 (where y; E R² Vi) must…

A: ℝ3 vector space is a three dimensional space. So, the basis for ℝ3 must have exactly three linearly…

Q: Consider the vectors from R 1 1 v, = 2 V2 a V3= 4 %3D 5 a) Create the system which correspond to the…

A:

Q: 3. Find the vector component of u along a and the vector component of u orthogonal to a for the…

A: Given that the vector u=2, 0, 1 and the vector a=1, 2, 3. We have to find the vector component of u…

Q: Q-IV) ---- Let u b. be vectors in R. (a) Verify that B (u, v, w) is an orthogonal basis for R. (b)…

A: Note: A set of vectors S is said to a basis of Rn if S consist of n linearly independent vectors in…

Q: If the scalar (dot product) of two unit vectors is zero, they are a) Linearly dependent b) Forming…

A: For the given statement we have to choose the correct option.

Q: 2. Let u e R™ and v E R be two non-zero vectors, in other words at least one component of the…

A: Given: Let us consider the u∈ℝm and v∈Rn are two non-zero vectors and assume A=uvT∈ℝm×n. a) Presume…

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.