Direction Angles

Pre-Calculus · Axiom Academy

LESSON Direction Angles of Vectors The direction angle is the single number that says which way a vector points — measured counterclockwise from the positive x-axis. 1. The Angle That Names the Direction Place the vector in standard position — tail at the origin. The direction angle is the angle you sweep counterclockwise from the positive x -axis until you reach the vector. Drop the vector onto the axes and its two legs are the components: the horizontal leg is a , the vertical leg is b . the legs are the components a and b so the tangent of the angle is rise over run For (a 3-4-5 vector in Quadrant I): 2. One Rule for All Four Quadrants Here is the catch: on a calculator only ever returns an angle between and . But a real direction angle runs the whole way from to . Watch a vector make a full turn below — the true angle keeps climbing past , , , while bare would keep snapping back. The fix is one reference value plus a quadrant correction. For , the reference value is . Since sits in Quadrant II, add : 3. Running It Backward: Polar to Components Once you have the direction angle, you can rebuild the components. Fix the length r and swing the vector out to angle . Its horizontal shadow on the x -axis is and its vertical shadow on the y -axis is — those projections are the components a and b . A vector of magnitude r = 6 points at (Quadrant II). Its components are: a is negative and b is positive — exactly what a Quadrant-II vector should look like.

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