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...
Find two unit vectors orthogonal to both 1 = (4, 2, 4) and 7 (0,3, 8). Enter the two vectors in the form <a,b,c>, separated by commas. All the components should be in exact form (i.e. no decimal entries allowed). unit vectors = Submit Question
26 or 28 or both 25 28, find the vector projection of u onto v. Then write u In ExcTe wo orthogonal vectors, one of which is projyu. Insum of two orthogonal vect as a sum of two 26. (3.-7),2, 6) 27, u(8, 5), v 28, 2, 8), v-(9,-3 29 and 30, find the interior angles of the triangle with 25 28, find the vector projection of u onto v. Then write u In ExcTe wo orthogonal vectors, one of...
6 Let y = and u Write y as the sum of two orthogonal vectors, one in Span (u) and one orthogonal to u. 5 7 y=y+z=( (Type an integer or simplified fraction for each matrix element. List the terms in the same order as they appear in the original list.)
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 the line in R3 that passes through the point P and is perpendicular to both of the vectors v and w. Find an equation for the line L in vector form. (iii) Find parametric equations for the line L.
please answer both (12(8 pts) Find parametric equations of the line through the point (2, -1,3) and perpendicular to the line with parametric equations 1-t,y 4- 2t and 3+ t and perpendicular to the line with parametric equations 3+t,y 2-t and z 3+2t. (13)(8 pts) Find the unit tangent vector (T(t) for the vector function r(t) - costi+3t j+ 2sin 2t k at the point where t 0 (12(8 pts) Find parametric equations of the line through the point (2,...
(b) (i) Given the function f(x,y)= xe' + 3x, find two unit vectors who are orthogonal to the gradient of f at the point P(3,0). (7 Marks) (ii) In which direction does f(x,y)= xey + 3x increase most rapidly at the point P(3,0)? What is the maximum rate of change at that point? (3 Marks)
(1) Let 7 =< 2,1,-2 > and 7 =< 1,2,3 >. Find two vectors and such that ✓ = 7+7, where is parallel to 7 and is orthogonal to 7.
Find two unit vectors orthogonal to both ü = (2, -2, -3) and (0,3,6). unit vectors
Question 1 (2+2+5 marks] (a) Find the angle between the vectors y =(4,0,3), v = (0,2,0). (b) Consider the subspace V (a plane) spanned by the vectors y, V. Find an orthonormal basis for the plane. (Hint: you may not need to use the full Gram-Schmidt process.) (c) Find the projection of the vector w=(1,2,3) onto the subspace Vin (b). Hence find w as a sum of two vectors wi+w, where w, is in V and w, is perpendicular to...