Vectors and Matrices
Aligned to N-VM.A.1 — Common Core State Standards for Mathematics.
What this lesson teaches
A vector has both a magnitude (size) and a direction, like a plane's velocity. You can add vectors by combining their parts. A matrix is a grid of numbers used to organize data and to perform operations such as solving systems of equations.
Worked examples
Example 1
Add the vectors (3, 1) and (2, 4) by adding matching parts: 3 + 2 is 5 and 1 + 4 is 5, giving the vector (5, 5).
Example 2
A plane flies at <200, 0> km/h and a wind blows at <30, 40> km/h. The ground velocity is the sum: <230, 40>. Its speed is √(230² + 40²) = √(52900 + 1600) = √54500, about 233 km/h — slightly faster than the plane's airspeed, and pushed off course to the north.
Example 3
Common mistake: adding vector magnitudes instead of components. A 3-unit vector plus a 4-unit vector does not always give 7. If they point the same way you get 7; if they point opposite ways you get 1; if they are perpendicular you get 5. Only the COMPONENTS add directly, and the magnitude is computed afterwards from the resulting components.
Practice questions
- Add the vectors (1, 2) and (4, 3).
- What two pieces of information does a vector carry?
- What is a matrix used to organize?
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
Vectores y matrices
Un vector tiene tanto una magnitud (tamaño) como una dirección, como la velocidad de un avión. Puedes sumar vectores combinando sus componentes. Una matriz es una cuadrícula de números que se usa para organizar datos y realizar operaciones como resolver sistemas de ecuaciones.
Ejemplo 1: Suma los vectores (3, 1) y (2, 4) sumando las partes correspondientes: 3 + 2 es 5 y 1 + 4 es 5, dando el vector (5, 5).
Ejemplo 2: Un avión vuela a <200, 0> km/h y el viento sopla a <30, 40> km/h. La velocidad respecto al suelo es la suma: <230, 40>. Su rapidez es √(230² + 40²) = √(52900 + 1600) = √54500, unos 233 km/h — algo más rápido que la velocidad del avión en el aire, y desviado hacia el norte.
Ejemplo 3: Error común: sumar magnitudes de vectores en vez de componentes. Un vector de 3 unidades más uno de 4 no siempre da 7. Si apuntan en la misma dirección da 7; si apuntan en sentidos opuestos da 1; si son perpendiculares da 5. Solo las COMPONENTES se suman directamente, y la magnitud se calcula después a partir de las componentes resultantes.
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