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Vectors: Magnitude, Direction and Unit Vector

A vector is a quantity having both magnitude and direction. If a = a1i + a2j + a3k, then its magnitude is |a| = √(a12 + a22 + a32), and a unit vector in its direction is a/|a| when a is not the zero vector.

In one line: Magnitude is the length; unit vector is the same direction with length 1.

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Understand it

  • A position vector locates a point with respect to the origin.
  • For point P(x, y, z), OP = xi + yj + zk.
  • Direction cosines are the cosines of the angles made by a vector with the positive x-, y-, and z-axes.
  • If l, m, n are direction cosines, then l2 + m2 + n2 = 1.

Example

For a = 2i - 3j + 6k, |a| = √(4 + 9 + 36) = 7. A unit vector along a is (2/7)i - (3/7)j + (6/7)k.

Formula and rule

Key rules: |a1i + a2j + a3k| = √(a12 + a22 + a32); unit vector along non-zero a is a/|a|; direction cosines satisfy l2 + m2 + n2 = 1.

The unit vector formula is not defined for the zero vector.

See it worked out

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

Worked example

Find the unit vector along AB where A(1, 2, 3) and B(3, 6, 2).

  1. 1AB = (3 - 1)i + (6 - 2)j + (2 - 3)k = 2i + 4j - k. |AB| = √(4 + 16 + 1) = √21.
  2. 2Unit vector = (2i + 4j - k)/√21.

Method you can reuse

Write the vector in component form; square and add components for magnitude; divide each component by magnitude for unit vector; for direction cosines, identify component divided by magnitude.

Visual tip

A diagram showing a directed line segment from O to P(x, y, z), with projections on coordinate axes, helps connect position vector and direction cosines.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Find the magnitude of a = 3i + 4j - 12k.

Think first, then show answer
|a| = √(32 + 42 + (-12)2) = √169 = 13.

Common wrong answer

Dividing by the sum of components instead of the magnitude, for example using 2 + 4 - 1 instead of √21.

Don't mix this up

Students often write only the magnitude and forget that a vector answer also needs direction or component form.

Exam tip

Before finalising an answer, check whether the question asks for magnitude, vector, unit vector, position vector, or direction cosines.

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 usually ask for magnitude, unit vector, position vector of a point, or direction cosines of a given vector.
  1. 1

    Enough for 1 mark

    A vector is a quantity having both magnitude and direction. If a = a1i + a2j + a3k, then its magnitude is |a| = √(a12 + a22 + a32), and a unit vector in its direction is a/|a| when a is not the zero vector.

  2. 2

    Add for 2 marks

    Example: For a = 2i - 3j + 6k, |a| = √(4 + 9 + 36) = 7. A unit vector along a is (2/7)i - (3/7)j + (6/7)k.

  3. 3

    Add for 3 marks

    A position vector locates a point with respect to the origin.

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