Final answer:
A vector parametric equation for a line given a point and direction can be expressed as r(t) = № + tD. The scalar t varies to represent every point on the line, and this equation is critical for analyzing relative motion problems in one or two dimensions.
Step-by-step explanation:
To provide a vector parametric equation for a line that passes through a given point, you need a point on the line (usually given as a position vector) and a direction vector (which shows the line’s direction). Assume the given point is represented by the position vector №, and the direction vector is D. The parametric equation of the line can be written as:
r(t) = № + tD
Here, t is a scalar parameter, and as it varies, r(t) represents every point on the line. When t is multiplied by the direction vector D, we get a new vector that is always parallel to the original direction vector, and this process is known as vector multiplication. By adding the initial position vector № to the result of this multiplication, we obtain the location of a point on the line at a specific value of t.
For example, if the given point through which the line passes is P(2, 3, 4) with position vector № = <2, 3, 4> and the direction vector is D = <1, 0, -1>, the parametric equation is:
r(t) = <2, 3, 4> + t<1, 0, -1>
The use of vector parametric equations is important for analyzing one-dimensional and two-dimensional relative motion problems by incorporating both the position and velocity vector equations.