C
CraftExam
medium difficulty8 min

Vector Addition and Scalar Multiplication

Vector addition combines vectors by triangle law or parallelogram law. Scalar multiplication changes the magnitude and possibly direction of a vector: if k is a scalar, then ka has magnitude |k||a| and direction same as a for k > 0, opposite for k < 0.

In one line: Add vectors component by component; multiply a scalar into every component.

Practice This Concept

Understand it

In component form, vectors are added by adding corresponding components. Addition is commutative and associative. Scalar multiplication distributes over vector addition and is used to form linear combinations such as ma + nb.

Example

If a = 2i - j + 3k and b = -i + 4j + k, then a + b = i + 3j + 4k and 3a = 6i - 3j + 9k.

Formula and rule

Component rule: if a = a1i + a2j + a3k and b = b1i + b2j + b3k, then a + b = (a1 + b1)i + (a2 + b2)j + (a3 + b3)k.

Properties: a + b = b + a; (a + b) + c = a + (b + c); k(a + b) = ka + kb.

See it worked out

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

Worked example

Let a = 3i - 2j + k and b = -i + 5j + 2k.

  1. 1Find 2a + 3b.
  2. 22a = 6i - 4j + 2k.
  3. 33b = -3i + 15j + 6k.
  4. 4Therefore 2a + 3b = (6 - 3)i + (-4 + 15)j + (2 + 6)k = 3i + 11j + 8k.

Method you can reuse

Write each vector in component form; multiply scalars first; combine like components; if a geometric diagram is given, mark head-to-tail or parallelogram diagonal; state the final vector clearly.

Sketch it in your answer

Use a triangle-law diagram with the tail of the second vector at the head of the first, and a parallelogram-law diagram with two coinitial vectors and the diagonal as resultant.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

If a = i + 2j and b = 3i - j, find 2a - b.

Think first, then show answer
2a - b = 2(i + 2j) - (3i - j) = 2i + 4j - 3i + j = -i + 5j.

Common wrong answer

Changing signs incorrectly in expressions like 2a - b, especially when b has negative components.

Don't mix this up

Students may add magnitudes directly instead of adding vectors component-wise or geometrically.

Exam tip

When a diagram is involved, first decide whether the triangle law or parallelogram law matches the given arrangement.

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 for resultants, component simplification, verification of vector identities, and use of geometric laws in diagrams.
  1. 1

    Enough for 1 mark

    Vector addition combines vectors by triangle law or parallelogram law. Scalar multiplication changes the magnitude and possibly direction of a vector: if k is a scalar, then ka has magnitude |k||a| and direction same as a for k > 0, opposite for k < 0.

  2. 2

    Add for 2 marks

    Example: If a = 2i - j + 3k and b = -i + 4j + k, then a + b = i + 3j + 4k and 3a = 6i - 3j + 9k.

  3. 3

    Add for 3 marks

    In component form, vectors are added by adding corresponding components.

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