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medium difficulty8 min

Scalar Product, Angle and Projection

The scalar product of two vectors a and b is a · b = |a||b|cos θ. It gives a scalar, not a vector. In component form, if a = a1i + a2j + a3k and b = b1i + b2j + b3k, then a · b = a1b1 + a2b2 + a3b3.

In one line: Dot product measures how much one vector goes along another; zero means no along-part for non-zero perpendicular vectors.

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Understand it

Dot product connects algebra with angle. If a · b = 0 and both vectors are non-zero, then the vectors are perpendicular. Projection of a on b is (a · b)/|b| as a scalar component in the direction of b.

Example

For a = 2i + j - 2k and b = i - 3j + k, a · b = 2(1) + 1(-3) + (-2)(1) = -3.

Formula and rule

Key formulas: a · b = |a||b|cos θ; component form a · b = a1b1 + a2b2 + a3b3; cos θ = (a · b)/(|a||b|); scalar projection of a on b = (a · b)/|b|.

Conditions: angle formula needs non-zero vectors.

See it worked out

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

Worked example

Find the angle between a = i + j and b = i - j.

  1. 1First, a · b = 1(1) + 1(-1) = 0.
  2. 2Also |a| = √2, |b| = √2.
  3. 3Using cos θ = (a · b)/(|a||b|), cos θ = 0/(√2√2) = 0.
  4. 4Hence θ = 90 degrees, so the vectors are perpendicular.

Method you can reuse

Compute a · b using components; compute magnitudes if angle or projection is needed; substitute in the correct formula; check whether the expected result is scalar, angle, projection length, or work.

Visual tip

A diagram showing two vectors with included angle θ and projection of one vector on the other helps distinguish dot product, angle, and projection.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

If a = i + 2j + 2k and b = 2i - j, find a · b.

Think first, then show answer
a · b = 1(2) + 2(-1) + 2(0) = 0. The non-zero vectors are perpendicular.

Common wrong answer

Using sin θ instead of cos θ for dot product, or writing i, j, k in the final dot product answer.

Don't mix this up

Students sometimes treat the dot product answer as a vector or forget that perpendicularity requires non-zero vectors.

Exam tip

Use component dot product first, then compare with |a||b|cos θ if angle is required.

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

Common forms include finding angle between vectors, checking perpendicularity, finding projection, and work done by a force.
  1. 1

    Enough for 1 mark

    The scalar product of two vectors a and b is a · b = |a||b|cos θ. It gives a scalar, not a vector. In component form, if a = a1i + a2j + a3k and b = b1i + b2j + b3k, then a · b = a1b1 + a2b2 + a3b3.

  2. 2

    Add for 2 marks

    Example: For a = 2i + j - 2k and b = i - 3j + k, a · b = 2(1) + 1(-3) + (-2)(1) = -3.

  3. 3

    Add for 3 marks

    Dot product connects algebra with angle.

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