Geometry

Properties of a Vector

Vector Decomposition

         

Find the magnitude of the vector v=(3,4)?\vec{v}=(-3,4)?

Point P lies on side BC and divides it internally in the ratio 1 : 4 in the equilateral triangle ABCABC below. Let AB=a,\vec{AB}=\vec{a}, AC=b,\vec{AC}=\vec{b}, AP=c.\vec{AP}=\vec{c}. Then c\vec{c} can be expressed as xa+yb.x\vec{a}+y\vec{b}. Which of the following is equal to x×yx\times y?

GG is the centroid of triangle OABOAB below. Let OA=a,OB=b,\vec{OA}=\vec{a}, \vec{OB}=\vec{b}, and GA=c\vec{GA}=\vec{c}. Then c\vec{c} can be expressed as xayb.x\vec{a}-y\vec{b}. Which of the following is equal to xyx - y?

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?

NOTE: "South of east" and "north of east" are terms used in bearing; imagine a pointer straight east, and the rotates some angle clockwise (that's south of east) or counterclockwise (that's north of east).

Let the position vectors of three points A,B,A, B, and CC be a,b,\vec{a}, \vec{b}, and c,\vec{c}, respectively. Then 2AB+3BCCA2\vec{AB}+3\vec{BC}-\vec{CA} can be expressed as xa+yb+zc.x\vec{a}+y\vec{b}+z\vec{c}. Calculate x2+y2+z2x^2+y^2+z^2.

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