Vector Addition & Scalar Multiplication

Statement

Vectors can be combined in two main ways: by addition and by scalar multiplication. If two vectors are written as \((x,y)\) and \((u,v)\), then their sum is found by adding their corresponding components:

\[ \begin{pmatrix}x \ y\end{pmatrix} + \begin{pmatrix}u \ v\end{pmatrix} = \begin{pmatrix}x+u \ y+v\end{pmatrix} \]

If a vector is multiplied by a scalar \(k\), both components are multiplied by \(k\):

\[ k\begin{pmatrix}x \ y\end{pmatrix} = \begin{pmatrix}kx \ ky\end{pmatrix} \]

Why it’s true

• Each vector is an ordered pair describing horizontal and vertical movement.
• Adding vectors means combining those movements, so x-components add together and y-components add together.
• Scalar multiplication scales the length of the vector without changing its direction.

Recipe (how to use it)

1. Write both vectors in column form \((x,y)\).
2. For addition, add x-components together and y-components together.
3. For scalar multiplication, multiply each component by the scalar \(k\).

Spotting it

Look for questions asking to “find the sum of two vectors” or “multiply a vector by a constant”. These keywords indicate the use of this rule.

Common pairings

• Vector addition is often followed by finding magnitude or direction of the result.
• Scalar multiplication is often used before forming unit vectors or linear combinations.

Mini examples

1. Given: \((3,2)+(4,5)\). Find: Result. Answer: \((7,7)\).
2. Given: \(2(3,-4)\). Find: Result. Answer: \((6,-8)\).

Pitfalls

• Adding unlike components (e.g., adding x to y by mistake).
• Forgetting to multiply both components by the scalar.
• Misinterpreting scalar multiplication as adding instead of scaling.

Exam strategy

• Always align components vertically when adding.
• Check your result: if multiplying by a negative scalar, the direction reverses.
• Simplify fractions or factor out common terms where possible.

Summary

Vector addition combines two movements into one. Scalar multiplication stretches or shrinks a vector without changing its direction. Together, these operations form the basis of vector algebra, used in mechanics, geometry, and coordinate problems.