Finding Vector Components
Aligned to N-VM.A.2 — Common Core State Standards for Mathematics.
What this lesson teaches
A vector from an initial point P to a terminal point Q has components found by subtracting coordinates: <Qx − Px, Qy − Py>. These give the horizontal and vertical change.
Worked examples
Example 1
From P(1, 2) to Q(4, 6), the components are <4 − 1, 6 − 2> = <3, 4>. Its magnitude is √(3^2 + 4^2) = √25 = 5.
Example 2
A force pulls with magnitude 10 at an angle of 30° above horizontal. Its components are <10·cos 30°, 10·sin 30°> = <8.66, 5>. Check: √(8.66² + 5²) = √(75 + 25) = 10 ✓. Splitting a vector into components is what lets you add forces that point in different directions.
Example 3
Common mistake: subtracting the points in the wrong order. The vector from P(1, 2) to Q(4, 6) is <4 − 1, 6 − 2> = <3, 4> — always TERMINAL minus INITIAL. Reversing it gives <−3, −4>, which has the same magnitude 5 but points the opposite way. Magnitude alone cannot catch the error, so fix the order before computing anything.
Practice questions
- Find the components of the vector from (0, 0) to (5, −2).
- Find the components of the vector from P(−3, 1) to Q(2, 4).
- Find the vector from (7, 3) to (1, 3) and its magnitude.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Hallar las componentes de un vector
Un vector que va de un punto inicial P a un punto terminal Q tiene componentes que se hallan restando coordenadas: <Qx − Px, Qy − Py>. Estas dan el cambio horizontal y vertical.
Ejemplo 1: De P(1, 2) a Q(4, 6), las componentes son <4 − 1, 6 − 2> = <3, 4>. Su magnitud es √(3^2 + 4^2) = √25 = 5.
Ejemplo 2: Una fuerza tira con magnitud 10 en un ángulo de 30° sobre la horizontal. Sus componentes son <10·cos 30°, 10·sen 30°> = <8.66, 5>. Comprueba: √(8.66² + 5²) = √(75 + 25) = 10 ✓. Descomponer un vector en componentes es lo que permite sumar fuerzas que apuntan en direcciones distintas.
Ejemplo 3: Error común: restar los puntos en el orden equivocado. El vector de P(1, 2) a Q(4, 6) es <4 − 1, 6 − 2> = <3, 4> — siempre FINAL menos INICIAL. Invertirlo da <−3, −4>, que tiene la misma magnitud 5 pero apunta al lado contrario. La magnitud por sí sola no detecta el error, así que arregla el orden antes de calcular nada.
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