Dismiss
Skip Navigation

Vector Equation of a Line

Identifies the position vector of every point along the line.

Atoms Practice
Estimated10 minsto complete
%
Progress
Practice Vector Equation of a Line
 
 
 
MEMORY METER
This indicates how strong in your memory this concept is
Practice
Progress
Estimated10 minsto complete
%
Practice Now
Turn In
Vector Direction, Equations, and Applications

Feel free to modify and personalize this study guide by clicking “Customize.”

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 P and the unit vector x^: _________________________

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

.

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

Hint: It establishes the equation cos2α+cos2β+cos2γ=P2x+P2y+P2zP2x+P2y+P2z=1

.

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 9,20
  2. What is the direction of 2,18
  3. What is the direction of 7,5

Identify the directional cosines associated with the given vector:

  1. P=75,30,102  
    1. cos α 
    2. cos β 
    3. cos γ =
  2. P=145,130,25.75 
    1. cos α 
    2. cos β 
    3. cos γ =
  3. P=220,300,175 
    1. cos α 
    2. cos β 
    3. cos γ =

.

Click here for answers.

.

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.

.

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

.

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

.

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. D=3,4,7+d3,3,2 and F=2,11,7+f2,11,7
  2. D=15,3,3+d3,11,8 and F=5,3,6+f1,4,6
.
Click here for answers.

.

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 vectorx,W=F×x=F(x) cos θ .

.

Magnetic Force

The magnetic force can be described using the __________________ of the ______________ vector and the ______________ vector: F=qv×B where F is the _______________, is the charge of the particle, v is the _______________, and B is the vector representing the magnetic field.

What is the metric base-unit of force? _______________

.

Torque

What is torque? __________________________________________________________

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

.

  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?
.
Click here for answers.

Explore More

Sign in to explore more, including practice questions and solutions for Vector Direction.
Please wait...
Please wait...