Adding and Scaling Vectors

The second day of The Treasure Walk handed every group two instructions — walk this vector, then walk that one — and asked the only question that matters: where do you end up, and what single instruction would have taken you there directly? That single instruction is the sum of the two vectors, and the walking is the definition: place the tail of $\vec{b}$ at the tip of $\vec{a}$, and $\vec{a}+\vec{b}$ runs from the start of the first walk to the end of the second. Tip to tail.

Subtraction asks a different question with the same picture: $\vec{a}-\vec{b}$ is the vector that corrects $\vec{b}$ into $\vec{a}$ — what you would still have to walk, having done $\vec{b}$, to end up where $\vec{a}$ goes.

Scalar multiplication stretches: $3\vec{v}$ is three of those walks in a row, $\frac{1}{2}\vec{v}$ is half of one, and $-\vec{v}$ is the walk taken backward — same magnitude, opposite direction. A scalar rescales an arrow; it never turns one.

Components do the bookkeeping

Geometry defines the operations; components make them fast. Add coordinate by coordinate, scale coordinate by coordinate:

$(3,-1,2)+(1,4,-2)=(4,3,0)3(1,4,-2)=(3,12,-6)$

Your group also put the tempting properties on trial rather than assuming them: is vector addition commutative? Walking $\vec{a}$ then $\vec{b}$ traces a different path from $\vec{b}$ then $\vec{a}$ — but lands on the same spot, the far corner of the same parallelogram. Commutative, associative, distributive over scalars: all hold, and each one is a picture before it is a rule.

Crossing a river

You swim at 2 m/s straight across a river; the current carries you 1.5 m/s downstream. Taking “across” as $(0,2)$ and “downstream” as $(1.5,0)$, your actual velocity is the sum $(1.5,2)$, with magnitude $\sqrt{1.5^2+2^2}=\sqrt{6.25}=2.5$ m/s, angled ${\tan }^{-1}\left(\frac{1.5}{2}\right)\approx 36.9°$ downstream of straight across. You never aimed in that direction — the sum did. Ferries, planes in wind, and forces on a beam all work exactly this way.

Adding and scaling are the whole grammar of vectors — everything in Equations of Lines and Equations of Planes is built from “start at a point, add multiples of a direction”. What addition and scaling cannot do is multiply two vectors together; that takes two new ideas, starting with The Dot Product. The warm-up section of Dot and Cross Product Practice includes a plane flying through wind that is worth doing before then.

Curriculum connection

C2.1

perform the operations of addition, subtraction, and scalar multiplication on vectors represented as directed line segments in two-space, and on vectors represented in Cartesian form in two-space and three-space

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C2.2

determine, through investigation with and without technology, some properties (e.g., commutative, associative, and distributive properties) of the operations of addition, subtraction, and scalar multiplication of vectors

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C2.3

solve problems involving the addition, subtraction, and scalar multiplication of vectors, including problems arising from real-world applications Sample problem: A plane on a heading of N 27° E has an air speed of 375 km/h. The wind is blowing from the south at 62 km/h. Determine the actual direction of travel of the plane and its ground speed.

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