For any vectors and , consider the following function of t: q(t) = ( + t ) · ( +

Question

For any vectors and , consider the following function of t: q(t) = ( + t ) · ( + t ) Explain why q(t) 0 for all real t. The dot product of a vector with itself is equivalent to the magnitude of the vector squared. The t2 in the result is never negative. The dot product of any two vectors is never negative. (b) Expand q(t) as a quadratic polynomial in t using the properties of dot product. || ||2 + 2t( · ) + t2|| ||2 || ||2 + t2|| ||2 || || + 2t( · ) + t|| || A 100 meter dash is run on a track in the direction of the vector = 2 + 7. The wind velocity is = 5 + km/hr. The rules say that a legal wind speed measured in the direction of the dash must not exceed 5 km/hr. Find the component of which is parallel to . Give an exact answer. + Find the speed of the wind in the direction of the track. Round your answer to two decimal places. km/hr Will the the race results be disqualified due to an illegal wind? no yes

Explanation / Answer

(v+tw).(v+tw)
=||V||2 +t2||w||2 + 2t(v.w)

which is option a)

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