Converting vector-form to cartesian-form line equations

Vector form

A line in vector form has the equation:

…where:

• is the position vector of any point on the line,
• is the position vector of a specific point on the line,
• is the direction vector of the line,
• is a scalar multiplier (changing it gives a new point on the line).

Cartesian form

Cartesian form is the standard equation of a line, written as:

(or one of its variants).

Gradient

We can find the gradient of the line from the vector of the vector form equation.

• The top value of the direction vector is the change in .
• The bottom value of the direction vector is the change in .

From this, we can use the formula for gradient:

Converting from vector to cartesian form

Once we have the gradient, we can substitute it into the cartesian form equation () to find .

Convert the vector equation to cartesian form

• The direction vector is
  • Change in
  • Change in
• Gradient
• gives us that is a point on the line.
• Substitute into to find :
• Answer:

Convert the vector equation to cartesian form

• The direction vector is
  • Change in
  • Change in
• Gradient
• gives us that is a point on the line.
• Substitute into to find :
• Answer:

Convert the vector equation to cartesian form

• The direction vector is
  • Change in
  • Change in
• Gradient
• gives us that is a point on the line.
• Substitute into to find :
• Answer: