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This problem involves finding the measures of the angles of a triangle with vertices A(-1, 0), B(2, 1), and C(1, -2). Using the coordinates of the vertices, we can calculate the lengths of the sides of the triangle and apply the cosine rule to determine the angles at each vertex. Calculating these angles provides insight into the geometric properties of the triangle formed by these points in a Cartesian plane.
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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. 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. x
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: y n u –u –n x
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. N We need the magnitude of the resultant a + w and the measure of angle theta. w a + w E a
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: N w a + w E a The new ground speed of the airplane is approximately 538.424 mph, and its new direction is about 6.465 degrees north of east.
