Question 1: Explain how vectors and mathematics are used in GPS technology.
Question 2: Given two vectors a and b, find:
a) when is the magnitude of a+b more than that of a-b?
b) when is the magnitude of a+b less than that of a-b?
c) when is the magnitude of a+b equal to that of a-b?
d) when is la+bl=lal+lbl?
e) what is the minimum value of la+bl?
Question 3: 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.
Question 4: Give examples of THREE vectors that are collinear and prove it.
Question 5: Given a=[1, -3, 6] and b=[4, -5, -2], find a vector c so that a*(bxc)=0. Explain what is the relationship between the vectors a, b, and c in this case, and why? VERIFY THIS.
Question 6: a, b, and c are vectors. Prove that a*(b+c)=a*b+a*c for all a, b, c ∈ R3
Question 7: DRAW DIAGRAMS to explain the answers to the following questions.
a) Is it possible to have pa(b)=0 ?
b) Is it possible to have pa(b) undefined?
c) Is it possible to have pa(b)= pb(a) ?
e) Explain why pa(c)=pa(pb(c))?
Question 8: The position of a point on a line is given by the equation s(t)=t^3-6t^2+9t-4, where s is measured in meters and t in seconds. After 2 seconds, is it moving toward or away from the origin? Show your work.
Question 9: Water is leaking from a trough at the rate of 0.8 L/s. The trough has a trapezoidal cross section, where the width at the bottom is 55cm, at the top 85cm, and the height is 25cm. The length of the trough is 3m. Find the rate at which the height is changing when the depth of water is 11cm. INCLUDE A DIAGRAM.
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