Q1) For the two vectors: A =3 ax -2 ay, B= ay + az find: 1) 2Ax – Bx 2) the angle between A and B 3) projection of B on A

Q: Given the points P, Q, and R, how do we know that the vectors PQ (cross product) PR is perpendicular…

A: Given that there are three pints in the plain P,Q and R.The cross product between PQ and PR is…

Q: 'he sum and difference of two vectors à and B are equal in magnitude. Prove lgebraically that the…

A: Given: A→+B→=A→-B→ The magnitude of sum of A →and B→ is , A→+B→=A2+B2+2ABcosθ The magnitude of…

Q: Show that the vectors i - j, j - k, k - i are parallel to a plane.

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Q: 02. Approve that the area of the parallelogram spanned by the vectors a= (2, -2, 1) and c= (-8, 8,…

A: Please refer the attached image for complete solution.

Q: Show that the lines L1 : x = 4 – t, y = 2t, z = 2+t and L2 : x = 1+ 4t, y = 3 – 8t, z = 5 – 4t are…

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Q: Find the area of the parallelogram that has the vectors u = ⟨3, −1, 5⟩ and v = ⟨2, −4, 1⟩ as…

A: Now

Q: 1) in triple dot of vectors |(A x B). C = |(CX A). B\, where the vectors A, B, and C are not same…

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Q: Find the area of the parallelogram determined by the vectors: u = (1,0,–3),v = (2,3,0)

A: By using formula  Area of parallelogram

Q: Show that the lines L₁ : x = 4-t, y = 2t, z = 1+ t and L₂ : x = 1 + 2t, y = 3 - 4t, z = 5 – 2t are…

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Q: What is the unit vector perpendicular aial yA n = i-j+ 3k, n2 = 2i + j + 3k to the vectors?

A: Given that:n1=i-j+3k,  n2=2i+j+3k

Q: When two nonzero vectors w are called perpendicular or orthogonal?

A: Let two nonzero vectors are u and v Then the scalar product of the two vectors is given as,…

Q: Q1. Given three vectors A = -a,+2a,+3a, B = 3a,+4a,+5a, and C = 2a, -2a,+7a,, compute : a) Angle…

A: Given the vector A=-ax+2ay+3azB=3ax+4ay+5azC=2ax-2ay+7az

Q: Q1, Find Parame tric equations for the line through the Point (0,1, 2) that is U =i +2} + 3k and ū=…

A: “Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: In general, show that the vectors V = ai + bj and W = -bi +aj are always perpen-dicular. Assume a…

A: Given vectors are v=ai+bj and w=-bi+aj, where a≠0 ,b≠0. We know that two vectors are perpendicular…

Q: Show that the lines L1 : x = 2 – t, y = 2t, z = 1 +t and L2 : x = 1 + 2t, y = 3 – 4t, z = 5 – 2t are…

A: To show that the lines L1: x=2-t, y=2t, z=1+t and L2: x=1+2t, y=3-4t, z=5-2t are parallel and to…

Q: Find the angle 0 between vectors u = 2i - jand v = 6i + 4j.

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Q: The unit vector perpendicular to the vectors n1= i-j+3k ,n2=2i+j+3k is a vector?

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Q: what is the vector equation of the line of intersection of the planes z=5 and x=9

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Q: Show that the lines L:a =4- t, y = 2t, z =1+t and L2:2 =1+ 21, y = 3 – 4t, z = 5 – 2t are parallel…

A: This is a problem of 3D analytical geometry and application of calculus, vector calculus.

Q: What is the angle measure between the vectors u = (-5, 8) and v = (3, 9)?

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Q: 02. Approve that the area of the parallelogram spanned by the vectors a- (2, -2, 1) and c= (-8, 8,…

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Q: Find the area of a parallelogram formed by the vectors PQ and PR, where P = (2, 5, 3), Q = (– 3, –…

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Q: Find the area of a parallelogram formed by the vectors PQ and PR, where P = (6, 6, 0), Q = (3, - 5,…

A: Let the parallelogram be PQRS, whose 3 vertices are P(6,6,0), Q(3,-5,6) & R(3,1,4). Then, these…

Q: What is the area of the parallelogram determined by the vectors v = (1, 3, 6) and w = (-1,0, 3)?

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Q: Find the vector equation for the line of intersection of the planes 5x + 2y – 5z = 2 and 5x + 4z = 4…

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Q: Find the vector equation for the line of intersection of the planes 3x−2y+2z=1 and 3x+4z=−4 r=⟨ ,…

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Q: Show that the lines L1 : x = 4 – t, y = 2t, z = :5+t and L2 : x = 1 + 4t, y = 3 – 8t, z = 5 – 4t are…

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Q: Find the area of the parallelogram that has the vectors as adjacent sides.u = (3, 2, −1), v = (1, 2,…

A: Now,

Q: * For the vectors v=16,8) and W=18,6), validate the Schwarz and Triangle mequaiity and fino the…

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Q: What is the area of the parallelogram formed by vectors A = 3ax -ay +2az and B = ax -2ay + 2az

A: What is the area of the parallelogram formed by vectors A = 3ax -ay +2az and B = ax -2ay + 2az

Q: Find the area of a parallelogram formed by the vectors PQ and PR, where P = (- 3, 4, 3), Q = (6, 6,…

A: .

Q: The inner(dot) product of two vectors and with lengths a and b. is given by a-b = abcos(y+2 T) ,…

A: The statement is true. The given statement is true because , Given a→.b→=abcosγ+2π But, cos…

Q: Find the area of the parallelogram with vectors a = 3p+ 2q, b= 2p-qwhere \pl = 4, lq] = 3, (p, q)…

A: To find the area of the parallelogram with the vectors a=3p+2q,b=2p-q where p=4,q=3,p,q=3π4.

Q: Show that the lines L1 : x = 4 – t, y = 2t, z = 2+t and L2 : x = 1+ 4t, y = 3 – 8t, z = 5 – 4t are…

A: We have, L1: x=4-t, y=2t, z=2+t and L2: x=1+4t, y=3-8t, z=5-4t

Q: Verify that the two planes are parallel. -x + 6y + 3z = 2 -** + 2y + z = 3 Find the normal vector,…

A: We will use that if Ax+By+Cz=d is equation of plan than (A,B,C) is normal vector as well as…

Q: Which of the vectors OP = (1,-1,5) and og = (2,-1,0) is not perpendicular to the line Li- - z.…

A: The equation of a line that passes through the point x0 , y0 , z0 in three dimensional space is…

Q: For two nonzero vectors u and v, if u v=0 then u and v are parallel.

A: Given that u and v are two nonzero vectors   Dot product of two nonzero vectors u.v=|u|.|v|…

Q: Verify the parallelogram law for vectors u and v in R": |u + v|l? + ||u – v||° = 2||u||? + 2||v||?

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Q: Q2. Approve that the area of the parallelogram spanned by the vectors a= (2, -2, 1) and c= (-8, 8,…

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Q: Find the area of a parallelogram formed by the vectors PQ and PR, where P = (– 6, – 1, 3), Q = (- 1,…

A: The corresponding position vectors with respect to origin are,…

Q: The vector from the origin to point A is given as 6ax – 2a, – 4az, and the unit vector directed from…

A: Given vector is A→=6,−2,−4 and B→=B23,−23,13 And B−A=10 We have to find the coordinates of the point…

Q: Show that the lines L1 : x =1 – t, y = 2t, z = 5+t and L2 : x 1+ 2t, y = 3 – 4t, z = 5 – 2t are…

A: Let's find.

Q: The vector from the origin to point A is given as 6ax – 2a, – 4a-, and t unit vector directed from…

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Q: Show that the indicated diagonal ofthe parallelogram determined by vectors u and v bisects the…

A: The given parallelogram is   Here, we are given |u|=|v| So, u+v is the required diagonal

Q: Let u and v be vectors. If their dot product is zero what is the angle between u and v (in degrees)?

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Q: Show that the lines L1 : x = 4 – t, y = 2t, z = 5+t and L2 : x = 1+ 2t, y = 3 – 4t, z = 5 – 2t are…

A: To show that the lines L1: x=4-t, y=2t, z=5+t and L2: x=1+2t, y=3-4t, z=5-2t are parallel and to…

Q: 20. Show that the pair of vectors ' and ¯J is perpendicular.

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Q: Find the acute angle between the two vectors A = 2a, +a, + 3a, and B = ax – 3ay + 2a, by using the…

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Q: Find the measure of the smallest nonnegative angle between the two vectors. Round approximate…

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