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ScientistGrades 10–12⏱ 20 min

Vectors

Drag two arrows on a grid. Add them tip to tail, subtract and scale them, and read off components, length and direction.

Standards: HSN-VM.A.1HSN-VM.A.2HSN-VM.B.4HSN-VM.B.5

1Wonder

A plane points north at 500 km/h, but the wind blows east. Where does it really go? To answer, we need numbers that have a direction.

2Explore

Vector lab

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Turn your phone sideways, or open this lesson on a tablet or computer. The explanation and quiz below work here.

uvu = ⟨3, 2⟩length 3.61 · direction 33.7°v = ⟨1, 3⟩length 3.16 · direction 71.6°u + v = ⟨4, 5⟩length 6.4 · direction 51.3°⟨3 + 1, 2 + 3⟩|u + v| = √(4² + 5²) = 6.4SillyChili
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Try this
  • Drag the tips of u and v. How are the components of u + v related to those of u and v?
  • Make u = ⟨3, 4⟩. What is its length?
  • Choose u − v. Where does the result point when u and v are the same?
  • Choose k · u and make k negative. What happens to the arrow?

3Explain

553°SillyChili
Step 1 of 5

A vector has size and direction

A vector is an arrow: its length (magnitude) tells how much, and its direction tells which way. This one has length 5 and points at 53° above the x-axis.

Working with vectors⟨a,b⟩+⟨c,d⟩=⟨a+c,  b+d⟩∣⟨a,b⟩∣=a2+b2\begin{aligned} \langle a, b\rangle + \langle c, d\rangle &= \langle a + c,\; b + d\rangle \\ |\langle a, b\rangle| &= \sqrt{a^2 + b^2} \end{aligned}

4Practice

Question 1 of 5

What are the components of this vector?

5Recap

  • A vector has a length and a direction. Write it with components ⟨a, b⟩.
  • Length: √(a² + b²).
  • Adding: tip to tail, or add the components.
  • Scaling by k stretches the vector; a negative k reverses it.
  • Velocities, forces and displacements are all vectors.