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Vector Equation of a Line

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Vector Direction, Equations, and Applications
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Vocabulary

Complete the chart.
Word Definition
Unit Vector __________________________________________________________________
___________ describes each of the x, y, and z components of the relevant vector
___________ describe a vector as the result of individual magnitudes and directions as measured from the axes, starting at the origin
Skew Lines __________________________________________________________________
Vector Equation __________________________________________________________________
___________ from a scientific standpoint is force multiplied by distance

Vector Direction

What is the equation for a vector? _________________

For the figure above:

Write the equation of the angle between \overrightarrow{P} and the unit vector \hat{x}: _________________________

What are the equations for the direction angles β and γ? _______________________    ______________________

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In your own words, describe the Pythagorean Property of Direction Cosines. __________________________________________________________________________

Hint: It establishes the equation \mbox{cos}^2 \alpha + \mbox{cos}^2 \beta + \mbox{cos}^2 \gamma = \frac{P^2_x + P^2_y + P^2_z} {P^2_x + P^2_y + P^2_z} = 1

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What is the vector direction in degrees of the following 2-dimensional vectors, assuming the positive x-axis is 0o ?

  1. What is the direction of \left \langle 9, 20 \right \rangle
  2. What is the direction of \left \langle 2, 18 \right \rangle
  3. What is the direction of \left \langle 7, 5 \right \rangle

Identify the directional cosines associated with the given vector:

  1. \overrightarrow{P} = \left \langle 75, 30, 102\right \rangle  
    1. cos α 
    2. cos β 
    3. cos γ =
  2. \overrightarrow{P} = \left \langle 145, 130, 25.75\right \rangle 
    1. cos α 
    2. cos β 
    3. cos γ =
  3. \overrightarrow{P} = \left \langle 220, 300, 175\right \rangle 
    1. cos α 
    2. cos β 
    3. cos γ =

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Vector Equation of a Line

What is the standard form of a line? ______________________

We can specify a particular line in space by requiring that the equations for the two intersecting planes be _____________________. In a previous section we developed one equation for a plane given by = - nxxnyynzz. In this case Qy Px QP = 0 and = 0 must __________________ for all points on the line passing through points and Q.

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If we already know the position vectors for two points on the line, \overrightarrow{p} and \overrightarrow{q} , we can use the method of vector subtraction to determine the equation of the vector, \overrightarrow{v} = \overrightarrow{p} - \overrightarrow{q} . Therefore,\overrightarrow{r} = \overrightarrow{p} + k (\overrightarrow{p} - \overrightarrow{q}) , where varies from -∞ to ∞.

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In general, a vector equation is any function that takes any one or more ____________ and returns a ____________.

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Write the vector equation of the line defined by the the following points:

  1. (-1, -1, 7) and (3, 11, 8)
  2. (1, -3, 2) and  (-5, 3, -1)
  3. (25, 17, 42) and  (-16, 12, 23)
Determine if the two vectors are skew lines or if they intersect each other.
  1. \overrightarrow{D} = \left \langle 3, 4, 7 \right \rangle + d\left \langle 3, 3, 2 \right \rangle and \overrightarrow{F} = \left \langle -2, 11, 7 \right \rangle + f\left \langle -2, 11, 7 \right \rangle
  2. \overrightarrow{D} = \left \langle 15, 3, -3 \right \rangle + d\left \langle 3, 11, 8 \right \rangle and \overrightarrow{F} = \left \langle 5, 3, 6 \right \rangle + f\left \langle 1, -4, 6 \right \rangle
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Vector Analysis Applications

Vectors can be used in many real-life situations.

Work

The work done by forces can be determined using the ________________ of the force and the displacement vector\overrightarrow{\triangle x}, W = \overrightarrow{F} \times \overrightarrow{\triangle x} = F(\triangle x) \ \mbox{cos} \ \theta .

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Magnetic Force

The magnetic force can be described using the __________________ of the ______________ vector and the ______________ vector: \overrightarrow{F} = q\overrightarrow{v} \times \overrightarrow{B} where \overrightarrow{F} is the _______________, is the charge of the particle, \overrightarrow{v} is the _______________, and \overrightarrow{B} is the vector representing the magnetic field.

What is the metric base-unit of force? _______________

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Torque

What is torque? __________________________________________________________

How do you find and describe the torque with vectors? __________________________________________________________

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  1. An fighter jet has a true airspeed of 1000 km/h due east. There is a cross wind blowing 60 degrees east of south at 100 km/h. Calculate the velocity of the jet relative to the ground.
  2. A bird is headed 40° east of north at 67 mph. A tail wind is blowing 45° west of south at 68 mph. Determine the direction of the bird.
  3. A fighter pilot with a mass of MA = 80 kg sits in the cock-pit with his back horizontal to the ground. His jet is moving vertically with acceleration a. If the acceleration due to gravity on the pilot is g = 9.8 ms-2, write a mathematical expression for and calculate the value of the reaction force R between the pilot and the back of his seat in the jet when: 
    1. a = 0
    2. a = 8 ms-2 upwards
    3. a = 8 ms-2 downwards.
  4. Several evil villians are holding Spiderman in place with ropes. If Flat-Nose Frankie is pulling at a constant force of 19 pounds, 44° to the horizontal, Green-Toe Gary is pulling at a constant force of 29 pounds along 31° to the horizontal, and Jelly-Knee Jennifyr is pulling at a constant force of 26 pounds at an angle of 33°, How hard and in what direction is Orange-Chin Oswald pulling if Spiderman is stuck in place?
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