Week 10 · Vectors

Thursday

Scalars and aiming upstream

// Rowing across the drift

⏱ about 20 min

Thursday: Scalars and Aiming Upstream

"If I row twice as hard, my rowing vector doubles," Comet says, at the chalkboard in the shed.

"Twice ⟨4, 0⟩ is ⟨8, 0⟩," Wren says. "Both components times two. That is a scalar multiple."

"What about the drift? I want to land at the post, straight across."

Nova projects the drift arrow, then flips it to point the other way. "Would you like a hint? Multiply by -1."

"Minus one flips the arrow," Wren says. "If I add ⟨0, -3⟩ to my rowing, the drift cancels."

Comet writes it out. "Row ⟨4, -3⟩. Add drift ⟨0, 3⟩. Path ⟨4, 0⟩. Straight across!"

"Aim upstream by exactly the drift," Wren says. "What do you notice about the length of your stroke?"

"Longer. √(16 + 9) is 5 squares of rowing to go 4 squares across."

Multiplying a vector by a scalar

A scalar is a plain number. Multiplying a vector by a scalar multiplies every component by that number.

2 × ⟨4, 0⟩ = ⟨8, 0⟩. The arrow points the same way and is twice as long.

½ × ⟨8, 0⟩ = ⟨4, 0⟩. The arrow points the same way and is half as long.

-1 × ⟨0, 3⟩ = ⟨0, -3⟩. The arrow is the same length and points the opposite way.

A positive scalar keeps the direction. A negative scalar reverses it. The magnitude is multiplied by the scalar's absolute value.

$The rowing arrow ⟨4, 0⟩, its double ⟨8, 0⟩, the drift ⟨0, 3⟩ and the flipped drift ⟨0, -3⟩.$

A solved problem: landing at the post

The drift is ⟨0, 3⟩. Comet wants a path straight across, ⟨4, 0⟩.
She needs rowing + drift = path, so rowing = path - drift.
rowing = ⟨4, 0⟩ - ⟨0, 3⟩ = ⟨4, -3⟩. Aim 3 squares upstream for every 4 across.
Check: ⟨4, -3⟩ + ⟨0, 3⟩ = ⟨4, 0⟩. The drift is cancelled.
The rowing magnitude is √(16 + 9) = 5, more effort than the 4 straight across.

Why components add: the proof from the picture

Here is why tip to tail and adding components always agree. Read it first, then put it in order below.

Let u = ⟨a, b⟩ and v = ⟨c, d⟩, both drawn from the origin.
Slide v so its tail sits on the tip of u, the point (a, b).
Moving c across and d up from (a, b) lands at (a + c, b + d).
The sum arrow runs from the origin to (a + c, b + d).
End point minus start point gives its components: ⟨a + c, b + d⟩.
So u + v = ⟨a + c, b + d⟩. Tip to tail and adding components are the same thing.

THE PROOF, IN ORDER

?Slide v so its tail sits at the tip of u, the point (a, b)
?The sum arrow runs from the origin to (a + c, b + d)
?So u + v = ⟨a + c, b + d⟩
?From (a, b), going c across and d up lands at (a + c, b + d)
?End minus start gives components ⟨a + c, b + d⟩
?Let u = ⟨a, b⟩ and v = ⟨c, d⟩ from the origin

WHY THIS EXERCISEThe proof is one slide and one subtraction. It is why the chalk walk and the arithmetic always matched.

SCALARS AND AIMING

Read the question.
Tap your answer.

$Arrows on a grid: v ⟨4, 0⟩, 2v ⟨8, 0⟩$ Comet rows twice as hard. What is 2 × ⟨4, 0⟩?

$Arrows on a grid: v ⟨8, 0⟩, 1/2v ⟨4, 0⟩$ Half effort on the doubled stroke: what is ½ × ⟨8, 0⟩?

$Arrows on a grid: v ⟨0, 3⟩, -1v ⟨0, -3⟩$ What is -1 × ⟨0, 3⟩, the vector that cancels the drift?

To land straight across with path ⟨4, 0⟩ against drift ⟨0, 3⟩, Comet rows path minus drift. What is her rowing vector?

WHICH SOLUTION MAKES SENSE?

Read the question.
Tap your answer.

The drift is ⟨0, 3⟩. Which rowing vector lands Comet straight across at ⟨4, 0⟩?

A vector is multiplied by -3. What happens to its arrow?

A plain number that multiplies a vector is called a this. Type one word.

The magnitude of ⟨4, -3⟩ is how many squares? Type the number.

Multiplying a vector by a negative scalar does what to its direction? Type one word.

Statement	True or false?
2 × 4 = 8	?
Multiplying by ½ halves the magnitude and keeps the direction.	?
Multiplying by -1 changes the magnitude.	?
Rowing ⟨4, -3⟩ against drift ⟨0, 3⟩ gives a path of ⟨4, 0⟩.	?
To cancel a drift you add the drift vector again.	?

WHY THIS EXERCISEAiming upstream is adding the negative of the drift, and that is the heart of the rowing problem.

Try it

Pick a drift of your own, like ⟨0, 2⟩ or ⟨1, 3⟩. Find the rowing vector that lands 6 squares straight across.
Check it by adding. Then find the rowing magnitude and compare it with 6.

Boathouse Log: write the three things a scalar can do to an arrow, with one example each.Write the aiming problem in your own words: what Comet knew, what she wanted, and the subtraction that found it.Write why the straight-across trip takes more rowing than the drifting trip did.

Sharp thinking. Tomorrow vectors leave the lake: paths across the lawn, a mixed review, and your log.

Grown-up checked my answers

← Wednesday Friday →