Let us imagine you are walking in a straight path in space. To describe your movement mathematically, we need a starting point and a direction. This is exactly what the vector equation of a line describes in mathematics.
In vector algebra, we use the vector equation of a line to describe lines in both 2-dimensional and 3-dimensional space. A vector is a quantity that has both magnitude (size) and direction. It is usually written in component form as:
v = (a, b), for 2D
Or,
v = (a, b, c), for 3D
Vectors represent direction and movement in space. When defining a line, we need:
A fixed point on the line (called the position vector)
A direction vector which gives the orientation of the line
A vector equation of a line is a representation of a straight line in 2D or 3D, which expresses all the points on the line using a position vector and a direction vector.
Equation
The vector equation of a line is given by:
r = r0 + td
Here,
r = (x, y, z) is the position vector of any point on the line
r0 = (x0, y0, z0) is the position vector of a known point on the line
d = (a, b, c) is the direction vector.
t is a scalar parameter that varies over all real numbers
This equation shows that as t increases or decreases, the vector r moves along the direction vector d and traces out all points on the line. A positive t moves forward, and a negative t moves backward along the line.
Note: The vector equation of a line can also be represented as:
r = (x0, y0, z0) + t(a, b, c)
⇒ (x, y, z) = (x0 + at, y0 + bt, z0 + ct)
Thus, x = x0 + at, y = y0 + bt, and z = z0 + ct
These equations are known as the parametric equations of a line that define the coordinates of any point on the line in terms of t.
Find the vector equation of a line passing through point A(1, 2) with a direction vector d = (3, -1).
Solution:
As we know, the vector equation formula: r = r0 + td Substituting r0 = (1, 2) and d = (3, -1) Here, r = (1, 2) + t(3, -1) ⇒ (x, y) = (1 + 3t, 2 – t), here t is a real number. Thus, the vector equation is: r = (1 + 3t, 2 – t)
Find the vector equation of a line passing through point A(2, -1, 4) with a direction vector d = (3, 5, -2)
Solution:
As we know, the vector equation formula: r = r0 + td Substituting r0 = (2, -1, 4) and d = (3, 5, -2) Here, r = (2, -1, 4) + t(3, 5, -2) ⇒ (x, y, z) = (2 + 3t, -1 + 5t, 4 – 2t), here t is a real number Thus, the vector equation is: r = (2 + 3t, -1 + 5t, 4 – 2t)
From Two Points
If two distinct points on the line are known, the vector equation of the line can be derived by first determining the direction vector and then applying the general equation of a line.
Let us consider two points A(x1, y1, z1) and B(x2, y2, z2).
The direction vector can be obtained by subtracting the coordinates of point A from those of point B.
d = B – A = (x2 – x1, y2 – y1, z2 – z1)
Thus, the general form of the vector equation of a line is:
r = A + t(B – A)
⇒ r = (x1, y1, z1) + t(x2 – x1, y2 – y1, z2 – z1)
Find the equation of a line passing through A(2, -1, 3) and B(5, 2, 6).
Solution:
Here, A = (2, -1, 3) B = (5, 2, 6) The direction vector is: d = B – A ⇒ d = (5 – 2, 2 – (-1), 6 – 3) ⇒ d = (3, 3, 3) As we know, the vector equation formula is: r = A + t(B – A) Here, substituting the values, ⇒ r = (2, -1, 3) + t(3, 3, 3) ⇒ r = (2 + 3t, -1 + 3t, 3 + 3t) Thus, the vector equation is: r = (2 + 3t, -1 + 3t, 3 + 3t)
Vector Equation vs. Cartesian Equation
Vector equations can be converted into Cartesian equations, also known as the symmetric form. This is done by eliminating the parameter t from the vector equations.
Here are the key differences between the two forms:
