811 8.4 Algebraically Defined Vectors and the Dot Product Algebraic Interpretation of Vectors A vector with initial point at the origin in a rectangular coordinate system is a position vector. A position vector u with endpoint at the point 1a, b2 is written 8a, b9, so u = 8a, b9. This means that every vector in the real plane corresponds to an ordered pair of real numbers. Thus, geometrically a vector is a directed line segment, and algebraically it is an ordered pair. The numbers a and b are the horizontal component and the vertical component, respectively, of vector u. Figure 33 shows the vector u = 8a, b9. The positive angle between the x-axis and a position vector is the direction angle for the vector. In Figure 33, u is the direction angle for vector u. The magnitude and direction angle of a vector are related to its horizontal and vertical components. 8.4 Algebraically Defined Vectors and the Dot Product ■ Algebraic Interpretation of Vectors ■ Operations with Vectors ■ The Dot Product and the Angle between Vectors y x b a u u = 〈a, b〉 0 (a, b) u Figure 33 LOOKING AHEAD TO CALCULUS In addition to two-dimensional vectors in a plane, calculus courses introduce three-dimensional vectors in space. The magnitude of the two-dimensional vector 8a, b9 is given by 2 a2 + b2. If we extend this to the threedimensional vector 8a, b, c9, the expression becomes 2 a2 + b2 + c2. Similar extensions are made for other concepts. Magnitude and Direction Angle of a Vector 8 a, b 9 The magnitude (length) of vector u = 8a, b9 is given by the following. ∣ u∣ =!a2 +b2 The direction angle u satisfies tan u = b a , where a≠0. EXAMPLE 1 Finding Magnitude and Direction Angle Find the magnitude and direction angle for u = 83, -29. ALGEBRAIC SOLUTION The magnitude is u = 232 + 1-222 = 213. To find the direction angle u, start with tan u = b a = -2 3 = - 2 3 . Vector u has a positive horizontal component and a negative vertical component, which places the position vector in quadrant IV. A calculator then gives tan-1 A - 2 3B ≈ -33.7°. Adding 360° yields the direction angle u ≈326.3°. See Figure 34. GRAPHING CALCULATOR SOLUTION The TI-84 Plus calculator can find the magnitude and direction angle using rectangular to polar conversion (which is covered in detail in the next chapter). An approximation for 213 is given, and the TI-84 Plus gives the direction angle with the least possible absolute value. We must add 360° to the given value -33.7° to obtain the positive direction angle u ≈326.3°. x y U 0 –2 2 3 u = k3, –2l u (3, –2) Figure 34 Figure 35 S Now Try Exercise 9.
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