Vector equations of lines

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.

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.

We can extend these kinematics concepts to two dimensions using vectors.

For example, the velocity vector ​v=(32t−1​)​ represents a constant velocity of ​3​ in the ​x​ direction, and a variable velocity ​vy​(t)=2t−1​ in the ​y​-direction.

The speed of the object is just the magnitude of the velocity:

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

which is just the equation of the scalar product.