Find a nonzero vector orthogonal to the plane through points P(2,1,5), Q(-1,2,3), and R(0,3,3), and find the area of  triangle PQR.

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

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Q: Find the vector v with the given length and the same direction as u.||v|| = 4, u = (−1, 1)

A: To find a vector v with the given magnitude of ||v|| = 4 and in the same direction as u = (−1, 1).

Q: Find two vectors that are orthogonal to the plane containing the points P = (4,-2,1) and R =…

A: We find two vectors that are orthogonal to the plane containing the points P = (3,0,1), Q =(4,-2,1)…

Q: Find an equation of the plane that passes through the point (1, 3, −7) and is perpendicular to the…

A: Direction ratio of the line n=j is (0,1,0)  Given that the line is perpendicular to the plane.  So…

Q: Find an equation of the plane that passes through the point (0, 0, 0) and is perpendicular to the…

A: The direction ratio of n = −3i + 2k is (-3,0,2) Given that that plane is perpendicular to n = −3i +…

Q: Given the vectors ū = (-2, 1, 1) and V = (3, –1, 1), calculate the vector projection of ū onto V

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Q: Find a vector parallel to the line of intersection of the planes 2x + 3y – z =11 and 4x + 2y – 3z =…

A: We need to find a vector parallel to the line of intersection of the planes 2x+3y-z=11 and 4x + 2y -…

Q: Find a nonzero vector parallel to the line of intersection of the two planes – (2x + 3y + z) = -2…

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Q: Consider the vectors a = (-1,–2, 3), b = (3,0, –2), v = (2, –5, –4) and the plane P given by 2x + y…

A: The given vector is; a→=-1,-2,3b→=3,0,-2v→=⟨2,-5,-4⟩and Plane P is;2x+y=z+13⇒2x+y-z=13…

Q: Given the vectors u = (2, -2, 3) and v = (-3, 2, – 1), calculate the vector projection of u onto v…

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Q: Find a vector that is orthogonal to the triangle with vertices P=(1,0,0), Q= (0,2,0), R=(1,1,1)

A: RECALL: Orthogonal vector:  Let us consider to vectors a→ and b→ then , cross product of a→ and b→…

Q: Find two unit vectors that are parallel to the yz-plane and are orthogonal to the vector 10i – 2j +…

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Q: Find the unit vector orthogonal to the plane through the points P ( 12, 3, -4 ), Q (-1, -2, 2), and…

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Q: Consider the points P(2, −1, 0), Q(5, 0, 0), and R(3, 1, √5). Find the coordinates of the vector QS…

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Q: Find a vector V that is perpendicular to the plane through the points A=(0,3,4) , B=(3,-5,−5) , and…

A: Find the vector AB and vector BC. The cross product of AB→ and BC→ gives a vector which is…

Q: Find a vector v that is perpendicular to the plane through the points А %3 (0, 4, —3), В 3 (5, —5,…

A: Coordinator of the plane is given Find out the vector that is perpendicular to the plane

Q: Find an equation of the plane that passes through the point P(-7,–5,5) and has the vector n = (-1,4,…

A: Given : Point P(-7,-5,5) and a normal vector n=<-1,4,2>   To Find : Equation of the plane that…

Q: Find the x-component of the vector of length 3 in the direction of V = (-1,-1).

A: v=<-1,-1>=-i-j Unit vector in the direction of vector V i.e.,     v⏜=-i-j(-1)2+(-1)2 =-i-j2…

Q: Consider the points P(3, x, -2) and Q(5, 3, 4) in R3 . Find x so that the vector from P to Q is…

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Q: Find an equation of the plane that passes through the point P(4, 6, 3) and has the vector n = (1,6,…

A: I have stated the used formula in the solution.

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: 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: Find the value of t such that the vectors v=(2t+1,-3,3t-2) and w=(7,4t+2,2) are orthogonal.

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Q: Find a vector v parallel to the intersection of the planes - (x + 3y + 4z) = 7 and 2y – 2x + 2z = 7.…

A: The direction vector of the line of intersection of two planes is parallel to the cross product of…

Q: M contains the point with coordinates (0,0,1) 2 M is orthogonal to the vector 0

A: Introduction: The general equation of Plane is ax-x1+by-y1+cz-z1=0

Q: Find the vector v with the given length and the same direction as u.||v|| = 3, u = (0, 2, 1, −1)

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Q: Find a vector v that is perpendicular to the plane through the points A = (4, – 5, –2), B = (2,3,…

A: We are asked to find the vector that perpendicular to the plane through the points

Q: Find a vector V of a magnitude equals 14 and perpendicular To the plane in space which includes the…

A: Explanation of the answer is as follows

Q: Find a vector v that is perpendicular to the plane containing 3 -() 6 and 0 2

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Q: Find a vector, v, in component form, that is orthogonal to the plane containing the points P = (4,…

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Q: Find the orthogonal projection of the vector (2,4) about the line L parallel to the vector (-1,2)

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Q: Find a vector x = ( c, d) that has dot products x · r = 1 and x · s = 0 with two given vectors r =…

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Q: Find a vector perpendicular to the 2-space line: Qx - Ny + R = 0

A: Please see the below picture for detailed solution.

Q: Find the length of the vector.v = (0, 1)

A: If Vector V = (V1,V2, V3) then the length of Vector ||V|| = (V1)2+(V2)2+(V3)2

Q: Find a vector ī that is perpendicular to the plane through the points A = (2, 5, 1), B = (5,4, –2),…

A: We find a vector v that is perpendicular to the plane through the points A = (2, 5, 1), B = (5,4,…

Q: Find a vector, in component form, that is orthogonal to the plane containing the points P = ( 5 , −…

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Q: Find two vectors in opposite directions that are orthogonal to the vector        u = ⟨3, 1, −2⟩

A: u=3, 1, -2 Let v=a, b, c be the vector orthogonal to the vector u=3, 1, -2. Then, u·v=0…

Q: Find a nonzero vector parallel to the line of intersection of the two planes 5y – 4x + z =-1 and –…

A: The line of intersection of two planes is orthogonal to their respective normals.

Q: Find an equation of the plane that passes through the point P(-3, -2, 3) and has the vector n = (-7,…

A: Find an equation of the plane that passes through the point P(-3, -2, 3) and has the vector n = (-7,…

Q: 5. Find a nonzero vector orthogonal to the plane through the points P(1,0, 1), Q(-2, 1, 3), and R(4,…

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Q: Find a vector, in component form, that is orthogonal to the plane containing the points P = ( – 1,…

A: find the explanation below

Q: Find a vector, in component form, that is orthogonal to the plane containing the points P = (5, 2,…

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Q: Find the vector equation for the line of intersection of the planes 4x + 2y + 2z -1 and 4x + z= -2…

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Q: Find the vector v with the given length and the same direction as u. ||v|| = 3, u = (1, 0, 1, –4)

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Q: perpendicular to the plane formed by the vectors w1 = [1 2 1] and w2 = [2 2 3], with the z axis

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Q: Find a nonzero vector parallel to the line of intersection of the two planes 5z – (3x+3y) : 2 and 2x…

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Q: Find a vector, v, in component form, that is orthogonal to the plane containing the points P = ( –…

A: Topic = Vector