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Trigonometry - Vectors

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Trigonometry - Vectors - Page 1 preview imageStudy GuideTrigonometry–Vectors1.The Rectangular Coordinate SystemFigure 1 Vectors drawn on a plane.In this chapter, we focus onvectors in a two-dimensional coordinate plane. The same ideas canbe extended to higher dimensions later, but for now, we’ll keep everything in 2D to make the conceptsclear.1.1Vectors and Standard PositionAvector𝐴𝐡⃗⃗⃗⃗⃗represents bothmagnitude (length)anddirection.If a vector𝐴𝐡⃗⃗⃗⃗⃗is moved so that itsstarting point is at the origin, it is said to be instandardposition.When a vector (𝑂𝑃⃗⃗⃗⃗⃗) starts at the origin and has thesame direction and magnitudeas (𝐴𝐡⃗⃗⃗⃗⃗), it iscalled thestandard vectorfor (𝐴𝐡⃗⃗⃗⃗⃗).You may also hear a standard vector called a:β€’position vectorβ€’radius vector
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Trigonometry - Vectors - Page 2 preview imageStudy Guide1.2Finding the Standard Vector Using CoordinatesTo find the standard vector for a vector drawn in the coordinate plane, we only need thecoordinatesof its endpoints.If:β€’A = (xa, ya)β€’B = (xb, yb)then the standard vector has coordinates:This works because the origin is at(0,0).Example1:Finding a Standard VectorFigure 2Drawing for Example 1.
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Trigonometry - Vectors - Page 3 preview imageStudy GuideGiven:A(-2,-7)and B(3, 2)Find the coordinates of point (P) so that𝑂𝐡⃗⃗⃗⃗⃗=𝐴𝐡⃗⃗⃗⃗⃗.Figure 3Components of a vector.Solution:So the standard vector is:1.3Algebraic Vectors and ComponentsAnalgebraic vectoris written as an ordered pair of real numbers:Here:β€’(a) is thehorizontal componentβ€’(b) is thevertical component
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Trigonometry - Vectors - Page 4 preview imageStudy GuideTwo vectors areequalif their corresponding components are equal.If both components are zero, the vector is called thezero vector.1.4Magnitude of a VectorThemagnitude(or length) of a vector⟨a, b⟩is found using the distance formula:Example2:Find the magnitude ofu =⟨3,-5⟩.Figure 4 Vector addition.1.5Vector AdditionTo add vectors, simplyadd their corresponding components.If:
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Trigonometry - Vectors - Page 5 preview imageStudy Guidethen:This matches the geometric idea of placing vectorstip-to-tail.1.6Scalar MultiplicationScalar multiplication means multiplying a vector by a number.If (q) is a scalar andv =⟨a, b⟩, then:This changes thelengthof the vector and may reverse itsdirection.Example3:Combining VectorsGiven:v =⟨8,-2⟩w =⟨3, 7⟩Find 5v-2w.1.7Unit VectorsAunit vectorhas a magnitude of1.To find a unit vector v in the same direction as a nonzero vector u:
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Trigonometry - Vectors - Page 6 preview imageStudy GuideExample4:Find a unit vector in the direction ofu =⟨7,-1⟩1.8The Unit Vectors i and jTwo special unit vectors are:Any vectorv =⟨a, b⟩can be written as:Example5:Writeu =⟨5, 3⟩using I and j.Figure 5 Drawing for Example 5.
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