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Question 5 (30 marks) Let A1, B(3,-5,0) and C(-1,4,1) be three points...
3 y+ z 0 2. Let W be a plane characterized by the equation W. D...
3 y+ z 0 2. Let W be a plane characterized by the equation W. D (5 Find an orthonormal basis for (57) Find the standard matrix for the orthogonal projection of R onto W 2) Find the distance between a vector (2, 2, 15) and the plane W. (5 (3
3 y+ z 0 2. Let W be a plane characterized by the equation W. D (5 Find an orthonormal basis for (57) Find the standard matrix for the...
5. (a) Let u 1,4,2), ,1,0). Find the orthogonal projection of u on v (b) Letu...
5. (a) Let u 1,4,2), ,1,0). Find the orthogonal projection of u on v (b) Letu ,1,0), u(0,1,1), (10,1). Find scalars c,,s such that 6. (a) Find the area of the triangle with vertices , (2,0,1), (3, 1,2). Find a vector orthogonal to the plane of the triangle. (b)) Find the distance between the point (1,5) and the line 2r -5y1 (i) Find the equation of the plane containing the points (1,2, 1), (2,1, 1), (1, 1,2). 7. (a) Let...
Q1. Given the points A: (0,0,2), B: (3,0,2), C: (1,2,1), and D: (2, 1,4 a) Find the cross product v - AB x AC. b) Find the equation of the plane P containing the triangle with vertices A, B, and C c)...
Q1. Given the points A: (0,0,2), B: (3,0,2), C: (1,2,1), and D: (2, 1,4 a) Find the cross product v - AB x AC. b) Find the equation of the plane P containing the triangle with vertices A, B, and C c) Find u the unit normal vector to P with direction v d) Find the component of AD over u and the angle between AD and u, then calculate the volume of the parallelepiped with edges AB, AC, AD...
72. Points and lines. Let L be the line passing through the points A (1,1,2) and...
72. Points and lines. Let L be the line passing through the points A (1,1,2) and B=(3,5,6). (a) Use a vector projection to find the closest point P on L to the point C = (2,1,1). (b) Find the distance between P and C.
Problem l: Let u, v and w be three vectors in R3 (a) Prove that wlv +lvlw bisects the angle betwe...
Problem l: Let u, v and w be three vectors in R3 (a) Prove that wlv +lvlw bisects the angle between v and w. (b) Consider the projection proj, w of w onto v, and then project this projection on u to get proju (proj, w). Is this necessarily equal to the projection proj, w of w on u? Prove or give a counterexample. (c) Find the volume of the parallelepiped with edges formed by u-(2,5,c), v (1,1,1) and w...
Let the vectors a = <1,2,3>, b= <1,1, 1 > and c = <1,2, 1 >...
Let the vectors a = <1,2,3>, b= <1,1, 1 > and c = <1,2, 1 > a) Determine whether the three are coplanar None of these 4 0.71 0.74 no b) Find the volume of the parallelepiped form c) Find the unit vector orthogonal to both ved d) Find the angle between the vectors a and 22.26 bunded to 2 decimal points) 12:21 e) Find the component of the vector a along 39.51 -1 ge
4 1|and b-l-2 Let A-13 a) Find the orthogonal projection p of b onto C(A) with its error vector. ...
4 1|and b-l-2 Let A-13 a) Find the orthogonal projection p of b onto C(A) with its error vector. b) Find the least squares approximation, £, to the solution vector x of Ai- c) The least squares error is defined to be the length of the vector b - AX. Find this vector and its length. d) What is the relationship between A, , and p?
4 1|and b-l-2 Let A-13 a) Find the orthogonal projection p of b onto...
1. Let Q = (-3.-3.-3.3), R = (-3.-3,-33) and S = (1,10,10.1). In the following, when...
1. Let Q = (-3.-3.-3.3), R = (-3.-3,-33) and S = (1,10,10.1). In the following, when rounding numbers, round to 4 decimal places. (i) Find QR and RS. (ii) Find the angle in degrees between QR and RS. (iii) Find ||QŘ|| and ||RŠI. (iv) Find the projection of R$ onto QR. 2. Let v = [6, 1, 2], w = [5,0,3), and P = (9,-7,31). (i) Find a vector u orthogonal to both v and w. (ii) Let L be...
1. Let Q = (-3, -3, -3.3), R = (-3, -3, -33) and S = (1,...
1. Let Q = (-3, -3, -3.3), R = (-3, -3, -33) and S = (1, 10, 10.1). In the following, when rounding numbers, round to 4 decimal places. (i) Find QŘ and RS. (ii) Find ||QR|| and ||RŠI. (iii) Find the angle in degrees between QR and RS. (iv) Find the projection of RŠ onto QŘ. 2. Let v= [6, 1, 2], w = [5,0,3], and P = (9, -7,31). (i) Find a vector u orthogonal to both v...
Let A-B-C denote B is between A and C. For this problem, use the following three...
Let A-B-C denote B is between A and C. For this problem, use the following three axioms: A1: Each pair of points is assigned a number, called the distance between A and B. it is denoted by AB. A2: Given any points A and B, then AB 20. Equality holds precisely when A=B. A3: For all points A and B, ABEBA Suppose A, B, C, and D are collinear. Assume that A-B-C means: A, B, and C are distinct, collinear...
3 y+ z 0 2. Let W be a plane characterized by the equation W. D (5 Find an orthonormal basis for (57) Find the standard matrix for the orthogonal projection of R onto W 2) Find the distance between a vector (2, 2, 15) and the plane W. (5 (3
3 y+ z 0 2. Let W be a plane characterized by the equation W. D (5 Find an orthonormal basis for (57) Find the standard matrix for the...
5. (a) Let u 1,4,2), ,1,0). Find the orthogonal projection of u on v (b) Letu ,1,0), u(0,1,1), (10,1). Find scalars c,,s such that 6. (a) Find the area of the triangle with vertices , (2,0,1), (3, 1,2). Find a vector orthogonal to the plane of the triangle. (b)) Find the distance between the point (1,5) and the line 2r -5y1 (i) Find the equation of the plane containing the points (1,2, 1), (2,1, 1), (1, 1,2). 7. (a) Let...
Q1. Given the points A: (0,0,2), B: (3,0,2), C: (1,2,1), and D: (2, 1,4 a) Find the cross product v - AB x AC. b) Find the equation of the plane P containing the triangle with vertices A, B, and C c) Find u the unit normal vector to P with direction v d) Find the component of AD over u and the angle between AD and u, then calculate the volume of the parallelepiped with edges AB, AC, AD...
72. Points and lines. Let L be the line passing through the points A (1,1,2) and B=(3,5,6). (a) Use a vector projection to find the closest point P on L to the point C = (2,1,1). (b) Find the distance between P and C.
Problem l: Let u, v and w be three vectors in R3 (a) Prove that wlv +lvlw bisects the angle between v and w. (b) Consider the projection proj, w of w onto v, and then project this projection on u to get proju (proj, w). Is this necessarily equal to the projection proj, w of w on u? Prove or give a counterexample. (c) Find the volume of the parallelepiped with edges formed by u-(2,5,c), v (1,1,1) and w...
Let the vectors a = <1,2,3>, b= <1,1, 1 > and c = <1,2, 1 > a) Determine whether the three are coplanar None of these 4 0.71 0.74 no b) Find the volume of the parallelepiped form c) Find the unit vector orthogonal to both ved d) Find the angle between the vectors a and 22.26 bunded to 2 decimal points) 12:21 e) Find the component of the vector a along 39.51 -1 ge
4 1|and b-l-2 Let A-13 a) Find the orthogonal projection p of b onto C(A) with its error vector. b) Find the least squares approximation, £, to the solution vector x of Ai- c) The least squares error is defined to be the length of the vector b - AX. Find this vector and its length. d) What is the relationship between A, , and p?
4 1|and b-l-2 Let A-13 a) Find the orthogonal projection p of b onto...
1. Let Q = (-3.-3.-3.3), R = (-3.-3,-33) and S = (1,10,10.1). In the following, when rounding numbers, round to 4 decimal places. (i) Find QR and RS. (ii) Find the angle in degrees between QR and RS. (iii) Find ||QŘ|| and ||RŠI. (iv) Find the projection of R$ onto QR. 2. Let v = [6, 1, 2], w = [5,0,3), and P = (9,-7,31). (i) Find a vector u orthogonal to both v and w. (ii) Let L be...
1. Let Q = (-3, -3, -3.3), R = (-3, -3, -33) and S = (1, 10, 10.1). In the following, when rounding numbers, round to 4 decimal places. (i) Find QŘ and RS. (ii) Find ||QR|| and ||RŠI. (iii) Find the angle in degrees between QR and RS. (iv) Find the projection of RŠ onto QŘ. 2. Let v= [6, 1, 2], w = [5,0,3], and P = (9, -7,31). (i) Find a vector u orthogonal to both v...
Let A-B-C denote B is between A and C. For this problem, use the following three axioms: A1: Each pair of points is assigned a number, called the distance between A and B. it is denoted by AB. A2: Given any points A and B, then AB 20. Equality holds precisely when A=B. A3: For all points A and B, ABEBA Suppose A, B, C, and D are collinear. Assume that A-B-C means: A, B, and C are distinct, collinear...
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