1. FroiI Show that cos a + cos B + cos y = 1 2. Use the figure from Ex. 1, to show that two nonzero vectors v and v2 in R' are orthogonal if and only if their direction cosines satisfy cos a, cos a, + cos B, cos B2 + cos Y1 cos Y2 = 0 3. Verify that the Cauchy-Schwarz inequality holds u = (0, 2, 2, 1), v (1,1, 1,1) %3D

Q: (3) Suppose a vector x is orthogonal to vectors u and v. Show that x is orthogonal to u+v.

A: We will use the fact that x is orthogonal to u and v to prove that x is orthogonal to u+v also

Q: 5. Let 0 be the angle between vectors ū = (1,3) and i = (5, 2). What is cos 0 and what is sin 0?

A: Angle between two vectors is given by,

Q: 3 Among all unit vectors u = y in R', find the one for which the sum x + 8y+ 4z is minimal.

A:

Q: 1. Let v = PQ where P = (3,4,0) and Q = (2, 3, 1) Find a vector u that is parallel to V but with a…

A:

Q: Let a E R" be fixed. Suppose that vectors x, y E R" are related by the equation a = x+ (x.y)y. a)…

A: Solutions

Q: Week 12: Part 1: This exercise will ask you to explore the equality Null(A") = (Col(A))-. Pick a…

A: Explanation-  For the given question, we will consider the standard unit vectors in ℝ4 Let us assume…

Q: The line through the points the points (xo,Yo, Zo) and (x1,y1, z1) be parametrized by the function T…

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Q: Find any two non collinear vectors on the plane x-y-3z +6=0

A:

Q: 5. If we have two vectors (V₁ and V₂), V₁ is a vector perpendic through the points A(1, 2, 5),…

A: Introduction: When a vector is along the direction in another vector, the two vectors are parallel…

Q: 14. Let nonzero vectors ü,ỷ,wER' be linearly independent. Show that there does not exist a nonzero…

A: We are given that three vectors in R3, u→,v→,w→ are linearly independent. We need to show, there…

Q: 2. The position vectors of A, B and C relative to a fixed origin O are respectively, Find, ABAC

A: [2].

Q: (a) Show that ||u + v||² = ||u||² + ||v||². (b) Deduce the Pythagorean theorem.

A: One has to prove the generalized Pythagorean Theorem.

Q: For the vectors v = ( 3, 4) and w = ( 4, 3) test the Schwarz inequality on v · w and the triangle…

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Q: 1. Let u and v be vectors in R". Suppose ||u||2 = 2, ||v||² = 1, and u·v = 1. Compute

A: