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5.Explain the meaning of direction angles and their relation to direction vectors. Solution.
a.What are the direction angles of the vector [-5, 1, 8]?
b.If a point P lies on the Solution x-axis, what are the direction angles of the position vectorOP? c. Prove that cos2α + cos2β + cos2γ = 1Solution.
d. A vector has direction angles α = 85° and β= 65°i.Find the value of γSolution.
ii.Find a vector that has those direction angles Solution. e.Explain why it is not possible for two of a vector's direction angles to be less than 45° Solution. f.What is the value of sin 2α +sin 2β+ sin2γ? Why? Solution.
6.Explain the meanings of the terms linearly dependent and coplanar. Make sure you demonstrate that you understand the difference between the terms, and the situation in which linear dependency implies coplanarity.
7.Determine if the vectors [2, 4, -1], [8, -10, 5] and [5, -3, 2] are coplanar.
8.Give examples of sets of three vectors that are a.
9.Explain how you would prove if four given points are coplanar. Use your method to determine if A (3, 4, -2), B (8, 5, 0), C (1, 10, -6) and D (9, 2, 2) are coplanar. Solution.
10.Determine if the following vectors are coplanar. Assume that v1, v2, v3 are not coplanar W1= 2v1 +7v2 W2= v2 + 2v3 W3= -v1 –7v3