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Properties of a Vector
Vectors allow you to represent quantities with both size and direction, such as the velocity of an airplane. Better yet, they do so in a mathematically-useful way. Dive in to see how!
Find the magnitude of the vector \(\vec{v}=(-3,4)?\)
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Point P lies on side BC and divides it internally in the ratio 1 : 4 in the equilateral triangle \(ABC\) below. Let \(\vec{AB}=\vec{a},\) \(\vec{AC}=\vec{b},\) \(\vec{AP}=\vec{c}.\) Then \(\vec{c}\) can be expressed as \(x\vec{a}+y\vec{b}.\) Which of the following is equal to \(x\times y\)?
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\(G\) is the centroid of triangle \(OAB\) below. Let \(\vec{OA}=\vec{a}, \vec{OB}=\vec{b},\) and \(\vec{GA}=\vec{c}\). Then \(\vec{c}\) can be expressed as \(x\vec{a}-y\vec{b}.\) Which of the following is equal to \(x - y\)?
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A commercial airliner can fly at a cruising velocity of 250m/s without the influence of the wind. If a 30m/s gust continuously blows 70\(^\circ\) south of east, then how many degrees north of east should the pilot angle the plane thereby counteracting the speed of the wind so that the plane flies directly east?
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Let the position vectors of three points \(A, B,\) and \(C\) be \(\vec{a}, \vec{b},\) and \(\vec{c},\) respectively. Then \(2\vec{AB}+3\vec{BC}-\vec{CA}\) can be expressed as \(x\vec{a}+y\vec{b}+z\vec{c}.\) Calculate \(x^2+y^2+z^2\).
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