- Identify a line in space as the intersection of two planes.
- Find points on a line given the vector equation for the line.
- Find the vector equation for a line given two points on the line.
You are probably very familiar with using y = mx + b, the slope-intercept form, as the equation of a line. While this equation works well in two-dimensional space, it is insufficient to completely define the equation of a line in higher order spaces. To help you see this clearly, several people have developed java-applets which allow one to rotate a vector and its coordinate system. Two of the best such applets can be accessed at http://www.univie.ac.at/future.media/moe/galerie/geom1/geom1.html#gb and http://www.walter-fendt.de/m14e/line3d.htm You may want to spend some time with one or both of these applets before reading the rest of this section.
Lines in Space
In a two-dimensional plane, a line can be represented by the equation y = mx + b, where m is the slope of the line and b is the y-intercept. In three-dimensional space, this equation can represent a line or a plane or the projection of a 3D line onto the x-y plane. For example, in the diagram below, the line on the left passes through points P and Q. The equation in 2D space for this line is y=−PQx+P
The three orange lines in the diagram at right below can also be described by this equation. All three have slope M=−PQ and y-intercept B = P. If we want to specify one of these lines in particular, we need to identify the line as the intersection of two planes. For example, the line where y=−PQx+P and z = 0 for all points is the intersection of the orange and green planes in the diagram below right. (Remember that the orange and green planes are drawn with edges for convenience, but that they represent planes that extend outward toward infinity.)
Therefore we can specify a particular line by requiring that the equations for both planes be simultaneously true. In a previous section we developed one equation for a plane given by d = -nx xo -ny yo -nz zo This means that in this case Qy + Px - QP = 0 and z = 0 must both be satisfied for all points on the line passing through points P and Q.
Finding the Vector Equation of a Line
Another way to identify points on a line can be found using the vector addition method we discussed earlier in this chapter. To find the position vector, r→, for any point along a line, we can add the position vector of a point on the line which we already know and add to that a vector, v→, that lies on the line as shown in the diagram below.
The position vector R→ for a point between P and Q is given by R→=p→+v→
All other points on this line can be reached by traveling along the line from point P, therefore the position vector for any point on the line is given by r→=p→+kv→.
If we know the locations of two points on a line, we can determine the equation of the line. All points on a line fulfill the equation r→=p→+kv→, where k is a scalar that varies from -∞ to ∞. 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→=p→−q→. Therefore, r→=p→+k(p→−q→), where k varies from -∞ to ∞.
Example: Write the equation of y=−53x+5 as a vector equation.
Solution: Start by choosing two points on the line, say (3,0) and (6,-5). Then p→=⟨3,0⟩ and q→=⟨6,−5⟩. So,
Finally, the vector equation of the line is
Example: Write the vector equation of the line defined by the points (2, 3, 4) and (-5, 2, -1).
Solution: These two points have position vectors ⟨2,3,4⟩ and ⟨−5,2,−1⟩. The vector of the line connecting the two points is given by
The equation of the line then becomes
Example 2: Determine the equation for the line defined by the points P = (6, 7, 5) and Q = (3, 2, 11). Then find the position vector for a point, R, half-way between these two points.
The equation of the line then becomes
Since the vector v→ points from P, the value of k for a particular point gives us some information about the location of that point. If 0 < k < 1, the point lies on the line between P and Q. If k< 0, the point is not between P and Q and is closer to P than Q. If k > 1, the point lies on the line between P and Q. If k < 0, the point is not between P and Q and is closer to P than Q. The point halfway between the two points has k=12.
Example: Air-craft control is tracking two planes in the vicinity of their airport. At a given moment, one plane is at a location 45 km east and 120 km north of the airport at an altitude of 7.5 km. The second plane is located 63 km east and 96 km south of the airport at an altitude of 6.0 km. The first plane is flying directly toward the airport while the second plane is continuing at a constant altitude with a heading defined by the vector h2−→=⟨3,4,0⟩ to land eventually at another airport to the northwest of our air-traffic controllers. Do the paths of these two aircraft cross? (Note: Two lines that are not parallel but do not cross are called skew lines.)
Solution: The first thing we need to do is to determine the position vectors of the two planes. Define our airport as the origin of coordinates and define x^ = east, y^ = north, and z^ = upward. Call the position of plane #1, P, and the position of plane #2, Q. This gives p→=⟨45,120,7.5⟩ and q→=⟨63,−96,6.0⟩
Plane #1 is heading directly toward the airport. The vector from the position of plane #1 to the origin is given by
The equation of the line representing plane #1’s motion then becomes
Plane #2 continues at a constant altitude with a heading h2−→=⟨3,4,0⟩. The equation of the line representing plane #2’s motion then becomes
If the paths of the two planes intersect, the position of that intersection must have coordinates which satisfy both equations.
To satisfy both equations, r1x must equal r2x, r1y must equal r2y, and r1z must equal r2z. This means that 45 - 45k1 = 63 + 3k2, 120 - 120k1 = -96 + 4k2, and 7.5 - 7.5k1 = 6.0 + 0k2 must all be simultaneously true.
The first equation simplifies to 15 - 15k1 = 21 + k2 or k2 = -6 - 15k1. The second simplifies to 30 - 30k1 = -24 + k2 or k2 = 54 - 30k1. The third equation simplifies to 7.5 - 7.5k1 = 6 or k1=15.
Inserting the value of k1 from the third equation into the first two equations, we find k2=−6−15(15)=−6−3=−9 and k2=54−30k1=54−30(15)=54−6=48 Since k2 = 33 and k2 = 0 cannot be simultaneously true, the two planes’ paths do not cross.
Note: There are other ways to reason through this problem. Can you find some? (Hint: Think of the geometry in 2-dimensions, or focus on the altitude as a way to simplify this problem).
General Vector Equations
In general, a vector equation is any function that takes any one or more variables and returns a vector. The vector equation of a line is an equation that identifies the position vector of every point along the line. This works for straight lines and for curves. For example, the vector equation p→=⟨3 cos θ,3 sin θ,2⟩ defines a circle having a radius of 3 units which sits parallel to the x-y plane at a distance of 2 units along the z-axis as shown below. The orange line is the set of points that satisfy the vector equation p→=⟨3 cos θ,3 sin θ,2⟩ while the grey lines are various individual position vectors which satisfy the vector equation.
In two-dimensional space, a line is defined by the equation y = mx + b where m is the slope of the line and b is the y-intercept. In three-dimensional space, a line is defined by the intersection of two planes. In other words, the line is the set of points for which the equations of the two planes are simultaneously true. Also, if we know the position vector p→ of one point on the line and the unit vector u^ in the direction of the line, all other points on the line satisfy the equation r→=p→+ku^. In general, a vector equation is any function that takes any one or more variables and returns a vector.
- Write the vector equation of the line defined by the points (5, -7, 3) and (2, 6, -4).
- Are the two vectors D→=⟨1,−1,4⟩+d⟨1,−1,1⟩ and F→=⟨2,4,7⟩+f⟨2,1,3⟩ skew lines or do they intersect one another?
- In physics, the motion of an object traveling at a constant speed is described by the equation st→=si→+v→t where si is the initial position, s is the position at some later time t, and v is the velocity of the object. Write the vector equation which returns the set of position vectors s→ for an object having an initial position si→=⟨2,3,4⟩ and a velocity of v→=⟨1,1,−2⟩ and determine the object’s location at t = 10s.
- An object has a position of si→=⟨3,3,6⟩ at t = 0 and a velocity of v→=⟨10,7,3⟩. Use the vector equation s→=si→+v→t to determine the distance traveled by the object between t = 3s and t = 5s. Distance measured in meters.
- Determine the vector equation of the straight line defined by the points (2, 2, 2) and (1, 3, 5).
- Do the two lines R→=⟨4,4,−2⟩+r⟨−3,7,2⟩ and K→=⟨9,−8,7⟩+k⟨−2,1,3⟩ intersect?