Unit 9 · Advanced TopicsN6 · Vectors · Propulsion

Vectors

Resolve components, add motion vectors, and find magnitude and direction.

28 min
Unlocks: Thrust Vectoring

Ready when you are

7 steps · about 28 minutes

Start with the warm-up.

Start

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Warm-up

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Key terms

vector: A quantity with magnitude and direction.
component: Horizontal or vertical part of a vector.
magnitude: The length of a vector.

Start by naming the quantity each problem asks for. That habit blocks the most common ACT trap.

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Try it first

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Take a calm first shot. Label what you know, write one relationship, and make the best choice you can.

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The big idea

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Build the picture before the arithmetic

Translate words into signed components: east and up positive, west and down negative. Add components first. Only then find the resultant magnitude or angle.

Key idea

The decision that matters

Vectors add component by component. Magnitude comes from the Pythagorean theorem.

Vector magnitude

\left\lVert\langle a,b\rangle\right\rVert=\sqrt{a^2+b^2}

Write this relationship before inserting numbers. It keeps labels and units attached to the work.

A compact example

A move 6 units east and 8 units north is the vector $\langle6,8\rangle$ , with magnitude $\sqrt{6^2+8^2}=10$ .

Cue	Best move
Combine motion	add signed components
Find length	square, add, and take the square root

Trap

Adding magnitudes instead of components

This error produces a believable answer. Pause before computing and say what each number means.

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Worked examples

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Skip check · 2 questions

Already know this? Get both right on the first try and you can skip the worked examples.

Worked example

Example 1: Add displacement vectors

Problem

A drone flies 7 kilometers east and 2 north, then 3 west and 6 north. Find its resultant vector and magnitude.

Use signed components: the vectors are

\langle7,2\rangle

and

\langle-3,6\rangle

Add components: $\langle4,8\rangle$ .

How should the two drone displacements be combined?

Magnitude is $\sqrt{4^2+8^2}=\sqrt{80}=4\sqrt5$ kilometers.

Why must components be added before finding magnitude?

Worked example

Example 2: Recover a component

Problem

A vector

\langle x,12\rangle

has magnitude 13 and points right. Find

x

Magnitude gives

\sqrt{x^2+12^2}=13

Square: $x^2+144=169$ , so $x^2=25$ .

Which root matches a vector that points right?

The vector points right, so $x$ is positive: $x=5$ .

Why does the direction select the positive root?

Faded example · you finish it

Example 3: You finish it

Problem

Add

\langle5,-2\rangle+\langle-1,7\rangle

Add horizontal components together and vertical components together.

Which completion is correct?

Which reason validates the result?

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Practice

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Spot the mistake

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

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Exit ticket

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.