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Vector Equation of a Line

Identifies the position vector of every point along the 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 \begin{align*}\overrightarrow{P}\end{align*}P and the unit vector \begin{align*}\hat{x}\end{align*}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 \begin{align*}\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\end{align*}cos2α+cos2β+cos2γ=P2x+P2y+P2zP2x+P2y+P2z=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 \begin{align*}\left \langle 9, 20 \right \rangle\end{align*}9,20
  2. What is the direction of \begin{align*}\left \langle 2, 18 \right \rangle\end{align*}2,18
  3. What is the direction of \begin{align*}\left \langle 7, 5 \right \rangle\end{align*}7,5

Identify the directional cosines associated with the given vector:

  1. \begin{align*}\overrightarrow{P} = \left \langle 75, 30, 102\right \rangle\end{align*}P=75,30,102  
    1. cos α 
    2. cos β 
    3. cos γ =
  2. \begin{align*}\overrightarrow{P} = \left \langle 145, 130, 25.75\right \rangle\end{align*}P=145,130,25.75 
    1. cos α 
    2. cos β 
    3. cos γ =
  3. \begin{align*}\overrightarrow{P} = \left \langle 220, 300, 175\right \rangle\end{align*}P=220,300,175 
    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, \begin{align*}\overrightarrow{p}\end{align*}p and \begin{align*}\overrightarrow{q}\end{align*}q , we can use the method of vector subtraction to determine the equation of the vector, \begin{align*}\overrightarrow{v} = \overrightarrow{p} - \overrightarrow{q}\end{align*}v=pq . Therefore,\begin{align*}\overrightarrow{r} = \overrightarrow{p} + k (\overrightarrow{p} - \overrightarrow{q})\end{align*}r=p+k(pq) , 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. \begin{align*}(-1, -1, 7)\end{align*} and \begin{align*}(3, 11, 8)\end{align*}
  2. \begin{align*}(1, -3, 2)\end{align*} and \begin{align*} (-5, 3, -1)\end{align*}
  3. \begin{align*}(25, 17, 42)\end{align*} and \begin{align*} (-16, 12, 23)\end{align*}
Determine if the two vectors are skew lines or if they intersect each other.
  1. \begin{align*}\overrightarrow{D} = \left \langle 3, 4, 7 \right \rangle + d\left \langle 3, 3, 2 \right \rangle\end{align*} and \begin{align*}\overrightarrow{F} = \left \langle -2, 11, 7 \right \rangle + f\left \langle -2, 11, 7 \right \rangle\end{align*}
  2. \begin{align*}\overrightarrow{D} = \left \langle 15, 3, -3 \right \rangle + d\left \langle 3, 11, 8 \right \rangle\end{align*} and \begin{align*}\overrightarrow{F} = \left \langle 5, 3, 6 \right \rangle + f\left \langle 1, -4, 6 \right \rangle\end{align*}
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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\begin{align*}\overrightarrow{\triangle x}, W = \overrightarrow{F} \times \overrightarrow{\triangle x} = F(\triangle x) \ \mbox{cos} \ \theta\end{align*} .

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Magnetic Force

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