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Vector Addition and Scalar Multiplication

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Student-friendly explanation

How to write this in exams

1
Start with the exact idea
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
Then show how to use it
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.
3
Add one concrete example
If a = 2i - j + 3k and b = -i + 4j + k, then a + b = i + 3j + 4k and 3a = 6i - 3j + 9k.
4
Avoid this incomplete answer
Changing signs incorrectly in expressions like 2a - b, especially when b has negative components.

Definition

Example

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

Rule to remember

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.

Memory hook

Add vectors component by component; multiply a scalar into every component.

Examples and method

Worked example

Let a = 3i - 2j + k and b = -i + 5j + 2k. Find 2a + 3b. 2a = 6i - 4j + 2k. 3b = -3i + 15j + 6k. Therefore 2a + 3b = (6 - 3)i + (-4 + 15)j + (2 + 6)k = 3i + 11j + 8k.

Method to apply

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.

Diagram support

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.

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

Common wrong answer

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

Exam tip

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

Quick check

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

2a - b = 2(i + 2j) - (3i - j) = 2i + 4j - 3i + j = -i + 5j.

Answer writing and exam use

1-mark answer

2-mark answer

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

3-mark answer

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. 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. Let a = 3i - 2j + k and b = -i + 5j + 2k. Find 2a + 3b. 2a = 6i - 4j + 2k. 3b = -3i + 15j + 6k. Therefore 2a + 3b = (6 - 3)i + (-4 + 15)j + (2 + 6)k = 3i + 11j + 8k. Questions ask for resultants, component simplification, verification of vector identities, and use of geometric laws in diagrams. Changing signs incorrectly in expressions like 2a - b, especially when b has negative components.

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