Greenwood press
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book-20600
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- Triangle Trigonometry
VECTORS
155 Vectors showing James j and Sarah s pulling on a wagon. Their resultant force is r, indicated by the diagonal of the parallelogram. Vector diagram for an airplane headed northeast that is blown off course by a wind 10° south of east. law of cosines r 2 = 400 2 + 90 2 − 2(400)(90) cos 125 ◦ . The length of r is about 458 miles per hour. The angle between a and r is about 9.3°. So the direction would be about 45 ◦ − 9.3 ◦ = 35.3 ◦ north of east. Even though the airplane would be pointed northeast, from the ground it would appear to be traveling only 35.3° north of east. (See Triangle Trigonometry.) When vectors are written as an ordered pair, the length is written first, and the angle second. (See Polar Coordinates.) Sarah’s vector would be written as s = [25, 20 ◦ ]; James’ vector would be j = [15, 0 ◦ ]. The brackets indicate that the vector is written in polar-coordinate form. The lengths of the vectors are written with the absolute value sign. The length of Sarah’s vector would be | s| = 25. Polar form is a natural way of presenting force vectors, but the algebra of vec- tors is easier to work with in Cartesian-coordinate form (x,y). This is called the component form. To convert a vector in polar form v = [d, θ] to component form, use the formulas x = d cos θ and y = d sin θ. Sarah’s polar vector would be s = (25 cos 20 ◦ , 25 sin 20 ◦ ) ≈ (23.50, 8.55).To reconstruct the length of Sarah’s vector from component form, use the Pythagorean theorem: | s| = (25 cos 20 ◦ ) 2 + (25 sin 20 ◦ ) 2 = 25. The addition of vectors in component form is done by the addition of coor- dinates. If v = (a, b) and w = (c, d), the parallelogram law requires that the vec- tor sum be v + w = (a + c, b + d). Component form makes it easier to handle problems involving gravity. If a golf ball is hit with an impact of 70 meters/sec- ond at a 30° angle, the distance of the ball (ignoring wind resistance and gravity) is given by the vector b = [70t, 30 ◦ ], where time t is given in seconds. The com- ponent form is b = (70t cos 30 ◦ , 70t sin 30 ◦ ). A graph would show the golf ball traveling upwards into space at an angle of 30° from the ground. However, grav- ity provides a force vector that reduces vertical distance as g = (0,–4.9t 2 ). The vector addition of the ball and gravity gives a parabolic path produced by b + g = (70t cos 30 ◦ , 70t sin 30 ◦ − 4.9t 2 ). Algebra can be used to determine how far the ball has traveled horizontally when it hits the ground. (See Angle for computations of the path of a projectile.) Vector descriptions of motions and forces are used to describe the collisions of atomic particles, the interaction of chemical substances, and the movements of stars and galaxies. Component form has operations that are somewhat like multiplication, but yet different. The dot product of two vectors is given by v • w = (ac, bd), where v = (a, b) and w = (c, d). Lengthening a vector by a scale factor k is given by k v = (ka, kb). The dot product is used in the formula for the cosine of an angle between two vectors: cos θ = v• w |v|| w| . The effectiveness of component-form vec- tors comes when vectors operate in more dimensions. For three-dimensional space, the dot product of v = (v 1 , v 2 , v 3 ) and w = (w 1 , w 2 , w 3 ) is v • w = (v 1 w 1 , v 2 w 2 , v 3 w 3 ), an easy-to-remember extension of the two-component model. Further, the equation for the cosine of the angle between two vectors looks exactly the same, even though there is an additional dimension. Download 1.81 Mb. Do'stlaringiz bilan baham: |
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