Adding and Subtracting Vectors
Aligned to N-VM.B.4 — Common Core State Standards for Mathematics.
What this lesson teaches
This is a (+) advanced standard. Add vectors component-wise: <a, b> + <c, d> = <a + c, b + d>. The magnitude of the sum is usually not the sum of the magnitudes. Scalar multiplication k<a, b> = <ka, kb> scales the length by |k|.
Worked examples
Example 1
Add <3, 4> + <1, −1> = <4, 3>. The magnitude of this sum is √(4^2 + 3^2) = √25 = 5, which is not 5 + √2 (the sum of the two lengths).
Example 2
Subtract <5, 2> − <1, 6> = <4, −4>. Its magnitude is √(16 + 16) = √32, about 5.66. Geometrically, this is the vector that points FROM the tip of <1, 6> TO the tip of <5, 2> — subtraction gives the displacement between two vectors.
Example 3
Common mistake: assuming subtraction commutes. <5, 2> − <1, 6> = <4, −4>, but <1, 6> − <5, 2> = <−4, 4> — same length, opposite direction, and in a physics problem that is the difference between a push and a pull. Vector addition commutes; vector subtraction does not, exactly as with ordinary numbers.
Practice questions
- Add <2, 5> + <3, −2>.
- Compute the scalar multiple 3<1, −4>.
- Find <6, 8> − <2, 5> 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
Sumar y restar vectores
Este es un estándar avanzado (+). Suma vectores componente por componente: <a, b> + <c, d> = <a + c, b + d>. La magnitud de la suma normalmente no es la suma de las magnitudes. La multiplicación por un escalar k<a, b> = <ka, kb> escala la longitud por |k|.
Ejemplo 1: Suma <3, 4> + <1, −1> = <4, 3>. La magnitud de esta suma es √(4^2 + 3^2) = √25 = 5, que no es 5 + √2 (la suma de las dos longitudes).
Ejemplo 2: Resta <5, 2> − <1, 6> = <4, −4>. Su magnitud es √(16 + 16) = √32, unos 5.66. Geométricamente, este es el vector que apunta DESDE la punta de <1, 6> HACIA la punta de <5, 2> — la resta da el desplazamiento entre dos vectores.
Ejemplo 3: Error común: suponer que la resta es conmutativa. <5, 2> − <1, 6> = <4, −4>, pero <1, 6> − <5, 2> = <−4, 4> — la misma longitud, dirección opuesta, y en un problema de física esa es la diferencia entre un empujón y un jalón. La suma de vectores sí es conmutativa; la resta no, igual que con los números ordinarios.
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