Equation of a Line in Vector and Cartesian Form

First identify a point on the line. Next find or read the direction ratios. If two points are given, subtract corresponding coordinates to get direction ratios. Write vector form using position vector plus λ times direction vector. Convert to Cartesian form by equating each coordinate expression to the same parameter.

A 3D axes diagram with one fixed point and a direction arrow supports the idea of generating all points on the line by changing λ.

For points A(1,2,-3) and B(3,1,1), writing direction ratios as (-2,1,-4) is acceptable only if used consistently, but mixing this with point B and signs from point A can produce a different line expression with errors.

A line in space is determined by one point on the line and one direction vector. Its vector form is r = a + λb, where a is the position vector of a fixed point and b is a direction vector.

A line in space is determined by one point on the line and one direction vector. Its vector form is r = a + λb, where a is the position vector of a fixed point and b is a direction vector. Vector form: r = a + λb. Cartesian form: (x - x₁)/a = (y - y₁)/b = (z - z₁)/c. Condition for use: (x₁, y₁, z₁) must be a point on the line and a, b, c must be direction ratios of the line. If any direction ratio is 0, the corresponding coordinate remains constant instead of being placed over 0 in ordinary symmetric form. The line through (1, -2, 3) with direction ratios 2, 1, -4 is r = i - 2j + 3k + λ(2i + j - 4k), or (x - 1)/2 = (y + 2)/1 = (z - 3)/(-4).

In vector form, r represents the position vector of any variable point on the line, a represents a known point on the line, b gives the direction, and λ is a real parameter. If the line passes through (x₁, y₁, z₁) and has direction ratios a, b, c, its Cartesian symmetric form is (x - x₁)/a = (y - y₁)/b = (z - z₁)/c, provided the relevant denominators are handled correctly. Vector form: r = a + λb. Cartesian form: (x - x₁)/a = (y - y₁)/b = (z - z₁)/c. Condition for use: (x₁, y₁, z₁) must be a point on the line and a, b, c must be direction ratios of the line. If any direction ratio is 0, the corresponding coordinate remains constant instead of being placed over 0 in ordinary symmetric form. Find the line through A(1, 2, -3) and B(3, 1, 1). Direction ratios from A to B are (3 - 1, 1 - 2, 1 - (-3)) = (2, -1, 4). Vector form: r = i + 2j - 3k + λ(2i - j + 4k). Cartesian form: (x - 1)/2 = (y - 2)/(-1) = (z + 3)/4. Questions ask students to form the equation from a point and direction, from two points, or convert between vector and Cartesian forms. Longer problems may use the line equation before finding angles or distances. For points A(1,2,-3) and B(3,1,1), writing direction ratios as (-2,1,-4) is acceptable only if used consistently, but mixing this with point B and signs from point A can produce a different line expression with errors.