That do not cross at right angles is the same as the angle determined by vectors normal

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That do not cross at right angles is the same as the angle determined by vectors normal to the lines or by the vectors parallel to the lines.image


Use this fact and the results of Exercise 31 or 32 to find the acute angles between the lines.


12x + 5y = 1, 2x - 2y = 3



Exercise 31


Show that v = ai + bj is perpendicular to the line ax + by = c by establishing that the slope of the vector v is the negative reciprocal of the slope of the given line.

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Thomas Calculus Early Transcendentals

ISBN: 9780321884077

13th Edition

Authors: Joel R Hass, Christopher E Heil, Maurice D Weir

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