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Finding Components
of Vectors

Learn to break any vector into its horizontal and vertical components by subtracting coordinates of the initial point from the terminal point.

Precalculus

What Is a Vector?

A vector is a quantity that has both magnitude (length) and direction. Unlike a plain number (a scalar), a vector tells you not just "how much" but also "which way."

Scalar magnitude only Vector magnitude + direction Initial Point Terminal Point

Position vs. Component Form

Position Form
Defined by two points: initial \(A(x_1, y_1)\) and terminal \(B(x_2, y_2)\).
Component Form
Written as \(\langle a, b \rangle\) where \(a\) is horizontal change and \(b\) is vertical change.

Why Components Matter

Breaking a vector into components lets you work with the horizontal and vertical parts separately. This is essential for adding vectors, computing magnitudes, and solving real-world physics and engineering problems.

Vector Addition
Add matching components: \(\langle a_1+a_2,\, b_1+b_2 \rangle\)
Magnitude
Use the formula \(\|\vec{v}\| = \sqrt{a^2 + b^2}\)
Direction
Find angle: \(\theta = \tan^{-1}\!\left(\frac{b}{a}\right)\)

The Component Formula

x y
\(A(1, 1)\)
\(B(5, 4)\)
\(5 - 1 = 4\)
\(4 - 1 = 3\)
\(\langle 4, 3 \rangle\)

Subtract initial from terminal: always terminal minus initial.

Breaking It Down Visually

Initial Terminal
horizontal: \(x_2 - x_1\)
vertical:
\(y_2 - y_1\)
vector \(\langle a, b \rangle\)

Vectors vs. Points

Vector \(\langle a, b \rangle\) Point \((x, y)\)
Notation Angle brackets \(\langle\;\rangle\) Parentheses \((\;)\)
Represents Displacement (change) Location (position)
Origin-dependent? No: direction + magnitude only Yes: fixed in the plane
Example \(\langle 3, -2 \rangle\) \((3, -2)\)

How to Find Components

1
Identify the points. Label the initial point \(A(x_1, y_1)\) and the terminal point \(B(x_2, y_2)\).
2
Subtract x-coordinates. Compute the horizontal component: \(a = x_2 - x_1\).
3
Subtract y-coordinates. Compute the vertical component: \(b = y_2 - y_1\).
4
Write in component form. The vector is \(\vec{AB} = \langle a,\, b \rangle\).

Example 1: Basic Components

Find the component form of \(\vec{AB}\) where \(A(2, 3)\) and \(B(7, 8)\).

\(A(2, 3)\)
\(B(7, 8)\)
\(7 - 2 = 5\)
\(8 - 3 = 5\)

Example 2: Negative Components

Find the component form of \(\vec{PQ}\) where \(P(6, 2)\) and \(Q(1, 7)\).

\(P(6, 2)\)
\(Q(1, 7)\)
\(1 - 6 = -5\)
\(7 - 2 = 5\)

Example 3: Both Negative

Find the component form of \(\vec{RS}\) where \(R(4, 9)\) and \(S(-2, 3)\).

1
Identify: Initial \(R(4, 9)\), terminal \(S(-2, 3)\).
2
Horizontal: \(x_2 - x_1 = -2 - 4 = \) \(-6\)
3
Vertical: \(y_2 - y_1 = 3 - 9 = \) \(-6\)
4
Result: \(\vec{RS} = \langle -6,\, -6 \rangle\). The vector points left and down.

Finding Magnitude from Components

Using Example 1: \(\vec{AB} = \langle 5, 5 \rangle\)

1
Square each component: \(5^2 = 25\) and \(5^2 = 25\)
2
Add: \(25 + 25 = 50\)
3
Take the square root: \(\|\vec{AB}\| = \sqrt{50} = 5\sqrt{2} \approx 7.07\)

Finding Direction from Components

x y
\(45^\circ\)
\(a = 5\)
\(b = 5\)
\(\langle 5, 5 \rangle\)

\(\theta = \tan^{-1}\!\left(\frac{5}{5}\right) = \tan^{-1}(1) = 45^\circ\)

Equivalent Vectors

Two vectors are equivalent if they have the same components, even if they start at different points. Only direction and magnitude matter.

A B C D
\(\langle 3, -2 \rangle = \langle 3, -2 \rangle\)
 Same components: equivalent vectors

Common Mistakes

Wrong Subtraction Order
Computing \(\langle x_1 - x_2,\, y_1 - y_2 \rangle\) instead of \(\langle x_2 - x_1,\, y_2 - y_1 \rangle\). This gives the opposite vector.
Mixing Coordinates
Using the \(x\) of one point with the \(y\) of the other: \(\langle x_2 - y_1,\, y_2 - x_1 \rangle\). Always subtract like with like.
Bracket Confusion
Writing the vector as \((a, b)\) with parentheses. Vectors use angle brackets \(\langle a, b \rangle\); parentheses denote points.

Quick Reference Summary

Concept Formula / Rule
Component Form \(\langle x_2 - x_1,\; y_2 - y_1 \rangle\)
Magnitude \(\|\vec{v}\| = \sqrt{a^2 + b^2}\)
Direction Angle \(\theta = \tan^{-1}(b/a)\)
Equivalent Vectors Same \(\langle a, b \rangle\), any start point
Zero Vector \(\langle 0, 0 \rangle\): no displacement

Question 1 of 5

Find the component form of \(\vec{AB}\) where \(A(3, 1)\) and \(B(8, 6)\).

Question 2 of 5

The vector \(\vec{PQ}\) has initial point \(P(7, -2)\) and terminal point \(Q(-1, 4)\). What is \(\vec{PQ}\)?

Question 3 of 5

A vector has components \(\langle -3, 4 \rangle\). What is its magnitude?

Question 4 of 5

Which two vectors are equivalent?

Question 5 of 5

A vector \(\vec{v} = \langle 6, 6 \rangle\). What direction angle does it make with the positive \(x\)-axis?