Q: Given three points in space P(1, -1, 0), Q(2, 1, -1),and R(-1, 1, 2), do the following: 1-Find the normal vector to the plane that contains all the three points. 2-Find the scalar product of PQ and PR and the angle 0 between them. 3-Show that |PQ x PR| = |PQ||PR| sin 0. 4-Prove that the area of a triangle PQR is PQ x PR|.

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: Q2: (a) Find the scalar and vector projection of ū along ủ, where u¯=(2, 3, 4) and v¯=(2, 2, 0). (b)…

A: Given : u=2,3,4, v=2,2,0 we know that scalar projection of u along v is given by…

Q: find T, N, and B for the curve at the indicated point. Hint: After finding T', plug in the specific…

A: The unit tangent vector T, the unit normal vector N and the unit binormal vector B are three…

Q: Q3: A given vectors r, = 2i - j+k, r=i+3j-2k, r, =-2i +j-3k and r=3i+2j+5k find scalars a, b and c…

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Q: Q1A)The vectors A = 3i+j+ 4k, B = 2i - 3j-k, find vector normal to A and In plane of A and, B. N

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Q: n Let ú and ú be two nonzero vectors in R" for which ú + 2v is orthogonal to ủ – 20. What is the…

A: We have to find the ratio

Q: For a plane with a normal of n = [1, 2, - 9]   and containing a point P (3, -4, 0), write the…

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Q: 10. Consider the plane P given by r + y – 3z = 0. (a) Find the projection of the vector (2, 1, 0)…

A: Given plane : x+y-3z=0  Projection of A along B is A.B/|B|

Q: Derive a parametric representation and find a normal vector for the plane 6x – 2y + 10z = 16. NOTE:…

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Q: Which of the following describes a plane perpendicular to the vector n = (2, 1, -4) and contains the…

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Q: Let P be the plane R3 with equation x+y-2z=4. the origin (0,0,0) is not in P! find two vectors in P0…

A: Given  x+y-2 z=4      .....(1)

Q: '. Given r(t)= (cos 21, sin 21, 31). Find the normal vector N(t), and explain. Then find the…

A: Given,  r(t)=(cos2t,sin2t,3t)             and point p(-1,0,3π/2)

Q: Q1A)The vectors A = 3i+j+ 4k, B = 2i3j-k, find vector normal to A and In plane of A and, B.…

A: Since you have posted multiple questions. As per guidelines we"ll solve only first first question.…

Q: The scalar component of H parallel to the line y=1, z=2 where H = -2a, + 3as- 4a, at p (1,90, 2) -3…

A: Given: H=-2aρ+3aϕ-4az p=1,90,2

Q: Let ū, and ūzare unique unit vectors normal to the plane containing a = 2i + 3j - k, and ,b =i-j+2k.…

A: Two vectors a and b in a plane are given. The vectors u1 and u2 are normal to the vectors a and b.…

Q: Q2] (A) Find the surface area of the portion of the half ball with radius a which cuts off by the…

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Q: Q The unit vector normal to the plane 2x + 3y+ 6z = 7 is (2ax +3ay +6 az)/7 (2ax - 3ay-6 az)/7 (4ax…

A: The given plane equation is 2x+3y+6z=7. The unit vector normal to the plane ax+by+cz=d is…

Q: Define a basis B = (ā,b, c) for R³, where -- 2 6= 1 3 Define P to be the plane spanned by d and b.…

A: To define a basis B=a→, b→, c→, we determine the determinant of matrix formed by vectors a→, b→, c→.…

Q: 24. Find v x w, where v and w are vectors of length 3 in the xz-plane, oriented as in Figure 16, and…

A: The vectors v and w are in the x-z plane. The cross product of both vectors will yield a vector…

Q: 2. Let each of the vector A = -8a, + 2ay – 4a,, B = 3ax – 4ay |3D Z> + 5a,, and C = 4ax + 5a, extend…

A: Using position vectors of given points. we ca find side vectors of triangle. using magnitude of a…

Q: 19. Supposethat the normalunit vector N of a surface S at z = f(x,y) is determined, how does the…

A: As per company guidelines we can solve first question. I hope that the given solution will help you.…

Q: Q1A)The vectors A = 3i+j+ 4k, B = 2i - 3j-k, find vector normal to A and In plane of A and, B.…

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Q: (a) Find a vector parallel to the line of intersection of the planes 4x – y – 3z = 6 and -5x + 4y –…

A: SOLUTION-

Q: Consider a plane II whose unit normal vector pointing outward from the origin is n = ai+ bj+ck and…

A: Projection of any vector x,y,z on the plane along the normal vector a,b,c will be d.

Q: Find v x w, where v and w are vectors of length 3 in the xz-plane, oriented as in the figure below,…

A: Explanation: Given that,  Vectors v and w length is 3 angle between vectors v and w:…

Q: The equation of the plane that passes through the point P(1,0,0) and has the vector n = (0,0, 1) as…

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Q: (B3) let V1, V2, . , Vn vectors in R". For all 1si, jsn, suppose that 1, si i= j, %3D 0, si i+ j,…

A: Let V(F) be a vector space. Let S=u1,u2,.........,un be a non empty subset of V. Then, the set is…

Q: 12. a. Use vectors to show that the distance from P;(x1, Y1, Z1) to the plane Ax + By + Cz = D is…

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Q: Q1. Prove that the points with  vectors a+2b, 2a-b, and 3a-4b are collinear

A: As per bartleby expert policy, only one individual question is to be answered. Kindly repost the…

Q: Q1): Find the value of a such that: 1. The vector V₁ = 3i + 2j + ak and V₂ = 2ai + 3j + ak are…

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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: (1) Let x and y be the vectors x = and y = 4 (la) Find a vector z such that ||z|| = 1 and z is…

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Q: a) Show that the line I = 0, y =t, :=t where teR a) lies in the plane 6r+4y-4r = 0 b) is parallel to…

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Q: (1) Let x and y be the vectors -- -- -) x = and y = 4 (la) Find a vector z such that ||z|| = 1 and z…

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Q: 1. a) Given the lines L₁ : 7 = (1,2,−4) + t(k+ 1,3k + 1, k − 3) and L₂: x=2-3s, y = 1-10s, z = 3-5s,…

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Q: (b) Two out-of-phase alternating voltages are represented by: V = 3.4sin (wt +=) and V, 1.3cos (wt…

A: The given problem is to find the sum of given vectors in the form of A sin(wt+ ϕ),we have to general…

Q: a) Find the distance between two skew lines L1 and L2 such that L1: 1+ 4t, y=5-4t, z=-1+5t  L2:…

A: Disclaimer: Since you have asked multiple questions, we will solve the first question for you. If…

Q: Let k ER and consider the plane T in R3 with vector equation X1 1 X2 k + s + t -1 s, t E R. X3 It is…

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Q: Given that F(x, y, z) = sin(yz)i + y²j+x³k, find (i) ▼ × F (ii) ▼·F and (iii) the component of V× F…

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Q: Find the value of d in the equation of the plane ax+by+cz+d=0 passing through the point P(3, 1, -2)…

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Q: Suppose that ū and i are vectors satisfying ủ ở = p , ||ủ|| = q , and 3 =. Find the angle (in…

A: (a) Consider the values for p, q and r and the values for the vector u, v and dot product for u and…

Q: Q3) Find the vector ž, which is perpendicular to vectors ã = (2,3, –1) and b = (1,–2,3) and…

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Q: 9. Find the normal unit vector n to the surface 62 + 2y²+ 2² = 225 at the point P: (5,5, 5).

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Q: 3) Prove for any two vectors à = (a,,a2) and b = (b,, b2) in R2 the angle between them is %3D d-b 0…

A: We use this formula to proof our result.

Q: In a three-dimensional space we have a vector R =( 4at + 3aj+ 5ak). Another vector S has the same Y…

A: 1st we will find other two vector R →  and T→ and then will use dot product and cross product to…

Q: Q1A)The vectors A = 3i+j+ 4k, B = 2i - 3j-k, find vector normal to A and In plane of A and, B. cos…

A: According to bartleby guidelines, we are supposed to do only one question. If you want more to be…

Q: 2. Determine the vector equation of the plane that contains the following two lines. L1: 7 = [ 4,…

A: