Vector Addition, Subtraction, and Scalar Multiplication Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Vector Addition, Subtraction, and Scalar Multiplication.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Vectors are added and subtracted component by component. Scalar multiplication multiplies each component of a vector by a number. If \mathbf{u} = \langle u_1, u_2 \rangle and \mathbf{v} = \langle v_1, v_2 \rangle, then \mathbf{u} + \mathbf{v} = \langle u_1 + v_1, u_2 + v_2 \rangle and k\mathbf{u} = \langle ku_1, ku_2 \rangle.

Vectors are arrows with direction and magnitude. Adding two vectors is like walking along the first arrow, then continuing along the second—you end up at the tip of the combined arrow (tip-to-tail method). Scalar multiplication stretches or shrinks the arrow: 2\mathbf{v} is twice as long in the same direction, while -\mathbf{v} points the opposite way.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Vector operations work component by component, preserving geometric meaning: addition chains displacements, and scalars change magnitude.

Common stuck point: Vectors must have the same number of components to be added or subtracted. A 2D vector cannot be added to a 3D vector.

Sense of Study hint: Line up the components vertically and add or subtract each pair separately, like column addition.

Worked Examples

Example 1

easy
If \mathbf{u} = \langle 3, -1 \rangle and \mathbf{v} = \langle 1, 4 \rangle, find 2\mathbf{u} - \mathbf{v}.

Solution

  1. 1
    Step 1: 2\mathbf{u} = \langle 6, -2 \rangle.
  2. 2
    Step 2: 2\mathbf{u} - \mathbf{v} = \langle 6 - 1, -2 - 4 \rangle = \langle 5, -6 \rangle.
  3. 3
    Check: Each component is 2u_i - v_i ✓

Answer

\langle 5, -6 \rangle
Scalar multiplication scales each component, then subtraction is done component-wise. Linear combinations of vectors like a\mathbf{u} + b\mathbf{v} are fundamental in linear algebra.

Example 2

medium
Find \mathbf{u} - \mathbf{v} where \mathbf{u} = \langle 2, 5, -1 \rangle and \mathbf{v} = \langle 4, -3, 2 \rangle.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Compute -3\langle 2, -4 \rangle.

Example 2

medium
Find the vector from point A(1, 3) to point B(4, -1).
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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