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So far we've worked with vectors that we think of as static objects. Now that we have a better understanding of their properties in general, we can translate more of our knowledge of scalar equations to vectors, starting with vector equations of a line.
Take a look at the animation below. It starts with a single copy of a vector v, then extends by adding on more copies of that same vector v. This creates a line with the same direction as the initial vector, with a magnitude given by however many copies of the initial vector we have made.
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The process demonstrated in the diagram above is essentially that of plotting a vector line.
Vector form
A vector line is defined by specifying one fixed point on the line and a direction vector. If the fixed point is given by position vector a, and the direction vector by b, then the line in vector form is:
Here, λ∈R is a parameter that varies over all real numbers, generating every point on the line. Changing λ moves the point r along the direction of b, creating the full infinite line.
We can also express this equation in terms of two points, P and Q, with position vectors p and q. Since two points uniquely determine a line, for any other point R on the line, the vector PR⃗ is in the same direction as PQ⃗, so you can write PR⃗=λPQ⃗ for some scalar λ. Using position vectors,
Example
Find the equation of the line passing through ⎝⎛101⎠⎞ parallel to ⎝⎛−123⎠⎞.
Discussion
The equation of a vector line gives both a position vector a and direction vector b. Why do you think this is the case? That is, why is it important we specify both a and b, instead of just giving one vector v and a parameter λ specifying how many "copies" we want to make of v?
A vector‐equation of a line is written as
where – a is a fixed “position” vector, picking out one point on the line; – b is a nonzero “direction” vector, showing which way the line goes.
If you tried to write a line as r=λv only, then as λ varies you can only reach points that are scalar multiples of v, i.e. that line must pass through the origin. But most lines in space do not pass through the origin, so you’d miss all of those.
Hence we need
• a to “shift” the whole line so it goes through the right point, and
• b to give its direction.
Checkpoint
Find the equation of the line passing through ⎝⎛2−43⎠⎞ parallel to ⎝⎛01−3/2⎠⎞.
Select the correct option
Exercise
The line r is given by the equation
Determine whether the point P(21,25,−23) is on the line.
If P is on the line, state the value of λ that gives its position vector.
Select the correct option
The form r=a+λb is a useful way to write the equation of a vector line due both to its brevity and clear focus on vectors. But sometimes, we may care more about how much the lines goes in different directions than where it points overall. In these cases we want to break r down into components based on its x-, y-, and z-coordinates.
Discussion
Let r be the line defined as
Can you split r up into equations for its x-, y-, and z-coordinates?
From
we read off each coordinate:
We call this way of expressing r parametric form.
Parametric form
A vector line in three-dimensional space can also be expressed in parametric form, showing explicitly the equations for the x, y, and z coordinates separately:
Here, (x0,y0,z0) is a point on the line, (l,m,n) are the components of the direction vector, and the parameter λ controls your position along the line. Changing λ moves the point continuously along the direction of the vector, producing the full infinite line.
Example
Express the line r=⎝⎛132⎠⎞+λ⎝⎛−320⎠⎞ in parametric form.
Checkpoint
Express the line r=⎝⎛5−28⎠⎞+λ⎝⎛111⎠⎞ in parametric form.
Select the correct option
Exercise
The line r is given by the parametric equations
Find the value of a for which the point P(5,5,8) is on r.
Select the correct option
Since x, y, and z have λ in common, we can also write the equation of a line by solving the equations of each coordinate for λ and setting them equal. This gives the Cartesian form of a vector line.
Cartesian form
The Cartesian form of a vector line in 3D is obtained by eliminating the parameter λ from the parametric form.
Solving each equation for λ gives:
Equating these expressions yields the Cartesian form:
which clearly emphasizes the consistent ratio of coordinate changes along the line.
If one of the direction vector components is zero, then that coordinate does not change as you move along the line.
Example
Express the line
in vector form.
Checkpoint
Express the line r=⎝⎛4−123⎠⎞+λ⎝⎛103⎠⎞ in Cartesian form.
Select the correct option
Exercise
Express the line
in vector form.
Select the correct option
One of the main reasons you might give the equation of a line as a vector, instead of the typical y= you're probably more used to, is because direction is central to the thing you're discussing. For example, we often use a vector line to represent motion, where knowing the direction (in three dimensions) that an object moves in is critical to understanding its behavior.
Let's work through an example to see what this means in practice.
Discussion
An object moves along a line with a constant velocity. It starts at the position (3,1,−1). Every second, it moves forward one unit in the x-direction, forward two units in the y-direction, and backward one unit in the z-direction.
Can you write equations for the motion of this particle in each of the three coordinate directions, giving a line in parametric form?
Now convert it to vector form, r=a+λb. When you find the vector value of r at a specific time λ, what are you finding specifically? That is, what does the value of r at a specific time λ tell you about the object?
Finally, what does the direction vector b represent in this context? What information about the object's movement does it provide?
Let t be the time in seconds. Since at t=0 the particle is at (3,1,−1) and each second it moves Δx=1, Δy=2, Δz=−1, its coordinates satisfy
In vector form we write r=a+λb. Here we may take λ=t, so
Evaluating r at a specific time λ=t gives the position vector of the object at that instant—that is, its (x,y,z)-coordinates. The direction vector b=(1,2,−1) represents the displacement per unit time (the velocity): it shows how far and in what direction the particle moves each second.
Modeling with vectors
In kinematics (the mathematical description of motion), 3D motion can be modeled by a vector line, often expressed with the parameter representing time, t.
Specifically, the direction vector b represents the velocity of an object, indicating both its direction and magnitude of movement. The magnitude of this vector, ∣b∣, is the object's speed—the rate at which it moves, irrespective of direction.
Checkpoint
An object moves with a constant velocity. Its position vector at time t seconds is given by
Find the speed of the object.
Select the correct option
Exercise
An object moves with a constant velocity. Its position vector at time t seconds is given by
Find the distance of the object from the origin when t=3 seconds.
Select the correct option
Another big reason to give a line's equation with vectors instead of scalars is that it's much easier to find angles when working with vectors.
Discussion
Two vector lines are given by the equations
Can you find the angle between the lines' direction vectors, ⎝⎛110⎠⎞ and ⎝⎛101⎠⎞, by using the scalar product formula cosθ=∣v∣∣w∣v⋅w?
Can you find the angle between the position vectors of r1 and r2 when λ,μ=1? When λ,μ=2?
Why aren't these angles the same? Which do you think best represents the angle between the lines themselves?
For the direction vectors v1=(1,1,0) and v2=(1,0,1) we have
When λ,μ=1
When λ,μ=2
The angles ϕ and ψ vary with λ because they measure the angle between the vectors from the origin to chosen points on each line. As you move the points along the lines, those position vectors change direction, so the angle depends on which points you pick. By contrast, direction vectors capture the lines’ fixed inclinations and do not change when you translate along a line. Hence the constant angle 60∘ between v1 and v2 is the true angle between the lines.
Angle between lines
The angle between two lines is simply the angle between their direction vectors.
For any two lines r1=a1+λb1 and r2=a2+μb2, the angle θ between r1 and r2 can be found via the formula
which is just the equation of the scalar product.
Example
Find the acute angle between the lines 3x−5=2y+2=z+1 and ⎝⎛13−5⎠⎞+λ⎝⎛312⎠⎞.
The direction vector of 2x−5=2y+2=z+1 is ⎝⎛321⎠⎞.
Both direction vectors have magnitude
so
Exercise
Find the acute angle between the lines r=⎝⎛−126⎠⎞+λ⎝⎛−2−12⎠⎞ and r=⎝⎛8−23⎠⎞+μ⎝⎛−632⎠⎞
Select the correct option
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