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Equations of a vector line
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Vector form, cartesian form, parametric form, modeling with vectors
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Vector form
AHL 3.14
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:
r =a+λb📖 =⎝⎛a1a2a3⎠⎞+λ⎝⎛b1b2b3⎠⎞
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.
Parametric form
AHL 3.14
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:
x=x0+λl,y=y0+λm,z=z0+λn📖
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.
Cartesian form
AHL 3.14
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:
λ=lx−x0,λ=my−y0,λ=nz−z0
Equating these expressions yields the Cartesian form:
lx−x0=my−y0=nz−z0📖
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.
Modeling with vectors
AHL 3.14
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.
Angle between lines
AHL 3.14
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 acute angle θ between r1 and r2 can be found via the formula
θ=cos−1(∣b1∣∣b2∣b1⋅b2)
which is just the equation of the scalar product.
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