medium difficulty10 min

Equation of a Line in Vector and Cartesian Form

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.

In one line: A space line needs a starting point and a direction arrow.

Practice This Concept

Understand it

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.

Example

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).

Formula and rule

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.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Find the line through A(1, 2, -3) and B(3, 1, 1).

1Direction ratios from A to B are (3 - 1, 1 - 2, 1 - (-3)) = (2, -1, 4).
2Vector form: r = i + 2j - 3k + λ(2i - j + 4k).
3Cartesian form: (x - 1)/2 = (y - 2)/(-1) = (z + 3)/4.

Method you can reuse

1First identify a point on the line.
2Next find or read the direction ratios.
3If two points are given, subtract corresponding coordinates to get direction ratios.
4Write vector form using position vector plus λ times direction vector.
5Convert to Cartesian form by equating each coordinate expression to the same parameter.

Visual tip

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.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Write the vector equation of the line through (2, 0, -1) and parallel to 3i - j + 2k.

Think first, then show answer

r = 2i - k + λ(3i - j + 2k), where λ is real.

Common wrong answer

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.

Don't mix this up

A common error is using the coordinates of the point as direction ratios or changing signs incorrectly while forming x - x₁, y - y₁, z - z₁.

Exam tip

Before writing the equation, underline the point and circle the direction vector or direction ratios. This prevents mixing their roles.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

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.

1
Enough for 1 mark
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.
2
Add for 2 marks
Example: 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).
3
Add for 3 marks
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.

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