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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).

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slide1

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:

slide2

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:

slide3

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:

slide4

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

Vector Applications

Section 10.2b

slide6

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

slide7

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

slide8

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:

slide9

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:

slide10

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

slide11

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:

slide12

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.