C
CraftExam
medium difficulty8 min

Zero, Unit, Equal, Parallel and Collinear Vectors

Vectors are classified by length, direction, and position. A zero vector has magnitude 0, a unit vector has magnitude 1, equal vectors have the same magnitude and direction, and collinear vectors are parallel to the same line.

In one line: Equal means same length and same direction; parallel means one is a scalar multiple of the other.

Practice This Concept

Understand it

  • Parallel vectors may have the same or opposite directions.
  • Coinitial vectors start from the same point.
  • Negative of vector a has the same magnitude as a but opposite direction.
  • These definitions are often tested through component comparison and geometric interpretation.

Example

Vectors a = 2i + 4j and b = i + 2j are parallel because a = 2b. Vectors 3i - j and 3i - j are equal if they have the same direction and magnitude, regardless of initial point.

Formula and rule

If a = λ b for some scalar λ, then a and b are parallel or collinear.

If λ > 0, same direction; if λ < 0, opposite direction.

Equal vectors require a = b component-wise.

See it worked out

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

Worked example

Check whether a = 6i + 9j - 3k and b = 2i + 3j - k are collinear.

  1. 1Compare ratios: 6/2 = 3, 9/3 = 3, (-3)/(-1) = 3.
  2. 2Since all ratios are equal, a = 3b.
  3. 3Therefore the vectors are collinear and have the same direction.

Method you can reuse

Identify components; compare magnitudes if needed; for equality compare each component; for parallel or collinear vectors check scalar multiple relation; state direction based on sign of scalar.

Visual tip

A diagram is useful but not essential; component comparison is usually enough for exam solutions.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Are a = 4i - 6j + 2k and b = -2i + 3j - k parallel?

Think first, then show answer
Yes. a = -2b, so they are parallel in opposite directions.

Common wrong answer

Calling two vectors equal because their magnitudes are equal, even when their directions differ.

Don't mix this up

Students confuse equal vectors with vectors drawn from the same initial point. Equal vectors need same magnitude and direction, not same location.

Exam tip

For component vectors, test parallel or collinear form by checking whether one vector is a scalar multiple of the other.

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

Short-answer questions may ask students to identify zero, unit, equal, parallel, opposite, or collinear vectors from diagrams or components.
  1. 1

    Enough for 1 mark

    Vectors are classified by length, direction, and position. A zero vector has magnitude 0, a unit vector has magnitude 1, equal vectors have the same magnitude and direction, and collinear vectors are parallel to the same line.

  2. 2

    Add for 2 marks

    Example: Vectors a = 2i + 4j and b = i + 2j are parallel because a = 2b. Vectors 3i - j and 3i - j are equal if they have the same direction and magnitude, regardless of initial point.

  3. 3

    Add for 3 marks

    Parallel vectors may have the same or opposite directions.

MCQ Quiz

Practice this concept with focused MCQs

Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.

10 MCQs5 MinutesInstant Results
Practice This Concept

Help improve this page

Found something confusing, incorrect, or missing?

Practice This Concept