Do Now: p.528, #27 Find the measures of the angles of the triangle whose vertices are A = (–1, 0), B = (2, 1), and C = (1, –2). Component forms: Magnitudes:

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Presentation transcript:

Do Now: p.528, #27 Find the measures of the angles of the triangle whose vertices are A = (–1, 0), B = (2, 1), and C = (1, –2). Component forms: Magnitudes:

Do Now: p.528, #27 Find the measures of the angles of the triangle whose vertices are A = (–1, 0), B = (2, 1), and C = (1, –2). Angle at A:

Do Now: p.528, #27 Find the measures of the angles of the triangle whose vertices are A = (–1, 0), B = (2, 1), and C = (1, –2). Angle at B:

Do Now: p.528, #27 Find the measures of the angles of the triangle whose vertices are A = (–1, 0), B = (2, 1), and C = (1, –2). Angle at C:

Vector Applications Section 10.2b

Suppose the motion of a particle in a plane is represented by parametric equations. The tangent line, suitably directed, models the direction of the motion at the point of tangency. x y A vector is tangent or normal to a curve at a point P if it is parallel or normal, respectively, to the line that is tangent to the curve at P.

Finding Vectors Tangent and Normal to a Curve Find unit vectors tangent and normal to the parametrized curve at the point where. Coordinates of the point: A graph of what we seek: x y u –u n –n

Finding Vectors Tangent and Normal to a Curve Find unit vectors tangent and normal to the parametrized curve at the point where. Tangent slope: A basic vector with a slope of 1/2: To find the unit vector, divide v by its magnitude:

Finding Vectors Tangent and Normal to a Curve Find unit vectors tangent and normal to the parametrized curve at the point where. The other unit vector: To find the normal vectors (with opposite reciprocal slopes), interchange components and change one of the signs:

Finding Ground Speed and Direction An airplane, flying due east at 500 mph in still air, encounters a 70-mph tail wind acting in the direction north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? Let a = airplane velocity and w = wind velocity. E N a w a + w We need the magnitude of the resultant a + w and the measure of angle theta.

Finding Ground Speed and Direction An airplane, flying due east at 500 mph in still air, encounters a 70-mph tail wind acting in the direction north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? Component forms of the vectors: The resultant: Magnitude:

Finding Ground Speed and Direction An airplane, flying due east at 500 mph in still air, encounters a 70-mph tail wind acting in the direction north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? Direction angle: E N a w a + w The new ground speed of the airplane is approximately mph, and its new direction is about degrees north of east.