Vector form, cartesian form, parametric form, modeling with vectors
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Practice exam-style equations of a line problems
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 βa, and the direction vector by βb, then the line in vector form is:
Here, βΞ»βRβ is a parameter that varies over all real numbers, generating every point on the line. Changing βΞ»β moves the point βrβ along the direction of βb, creating the full infinite line.
A vector line in three-dimensional space can also be expressed in parametric form, showing explicitly the equations for the βx, βy, and βzβ coordinates separately:
Here, β(x0β,y0β,z0β)β is a point on the line, β(l,m,n)β are the components of the direction vector, and the parameter βΞ»β controls your position along the line. Changing βΞ»β 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, βt.
Specifically, the direction vector βbβ represents the velocity of an object, indicating both its direction and magnitude of movement. The magnitude of this vector, ββ£bβ£, is the object's speedβthe rate at which it moves, irrespective of direction.
The angle between two lines is simply the angle between their direction vectors.
For any two lines βr1β=a1β+Ξ»b1ββ and βr2β=a2β+ΞΌb2β, the angle βΞΈβ between βr1ββ and βr2ββ can be found via the formula
which is just the equation of the scalar product.