Vector Addition, Subtraction, and Scalar Multiplication Formula

The Formula

\mathbf{u} + \mathbf{v} = \langle u_1 + v_1, u_2 + v_2 \rangle, k\mathbf{u} = \langle ku_1, ku_2 \rangle

When to use: 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.

Quick Example

\mathbf{u} = \langle 3, 1 \rangle, \quad \mathbf{v} = \langle -1, 4 \rangle
\mathbf{u} + \mathbf{v} = \langle 2, 5 \rangle, \quad \mathbf{u} - \mathbf{v} = \langle 4, -3 \rangle, \quad 2\mathbf{u} = \langle 6, 2 \rangle

Notation

Vectors in boldface (\mathbf{v}) or with an arrow (\vec{v}). Components in angle brackets \langle v_1, v_2 \rangle or column form. k is a scalar (number).

What This Formula Means

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.

Formal View

The vector space \mathbb{R}^n has operations: \mathbf{u} + \mathbf{v} = (u_1+v_1, \ldots, u_n+v_n) and k\mathbf{u} = (ku_1, \ldots, ku_n) for k \in \mathbb{R}. These satisfy the vector space axioms: commutativity, associativity, zero vector \mathbf{0}, and additive inverses.

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.

Common Mistakes

  • Adding vectors of different dimensions
  • Confusing vector addition with dot product—addition gives a vector, dot product gives a number
  • Forgetting that \mathbf{u} - \mathbf{v} means \mathbf{u} + (-\mathbf{v}), which points from \mathbf{v}'s tip to \mathbf{u}'s tip

Why This Formula Matters

Vectors model forces, velocities, and directions in physics. In computer science, they represent data points, image features, and word embeddings. Vector operations are the foundation of linear algebra.

Frequently Asked Questions

What is the Vector Addition, Subtraction, and Scalar Multiplication formula?

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.

How do you use the Vector Addition, Subtraction, and Scalar Multiplication formula?

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.

What do the symbols mean in the Vector Addition, Subtraction, and Scalar Multiplication formula?

Vectors in boldface (\mathbf{v}) or with an arrow (\vec{v}). Components in angle brackets \langle v_1, v_2 \rangle or column form. k is a scalar (number).

Why is the Vector Addition, Subtraction, and Scalar Multiplication formula important in Math?

Vectors model forces, velocities, and directions in physics. In computer science, they represent data points, image features, and word embeddings. Vector operations are the foundation of linear algebra.

What do students get wrong about Vector Addition, Subtraction, and Scalar Multiplication?

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

What should I learn before the Vector Addition, Subtraction, and Scalar Multiplication formula?

Before studying the Vector Addition, Subtraction, and Scalar Multiplication formula, you should understand: coordinate plane, expressions.

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