Components from magnitude and direction; unit vectors

About this lesson

Precalculus › Vectors

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Components from magnitude and direction; unit vectors

Warm-up

Three quick questions from earlier lessons.

Discover

A vector has length 10 and direction 30∘30^\circ. Its horizontal component is the side next to the 30∘30^\circ angle. Which is it?

  1. 10cos⁡30∘10\cos 30^\circ
  2. 10sin⁡30∘10\sin 30^\circ

Key idea

Magnitude and direction give the components, and a unit vector (length 1) gives just the direction.

A vector with magnitude rr and direction θ\theta is ⟨rcos⁡θ, rsin⁡θ⟩\langle r\cos\theta,\ r\sin\theta \rangle. The unit vector in the direction of v⃗\vec{v} is v⃗∣v⃗∣\dfrac{\vec{v}}{|\vec{v}|}.

Worked example

Write the vector with magnitude 6 and direction 120∘120^\circ.

6cos⁡120∘=−36\cos 120^\circ = -3
cos 120° = −1/2.
6sin⁡120∘=336\sin 120^\circ = 3\sqrt{3}
sin 120° = √3/2.
v⃗=⟨−3,33⟩\vec{v} = \langle -3, 3\sqrt{3} \rangle
Components.

Your turn, guided

Your turn: the unit vector in the direction of ⟨6,8⟩\langle 6, 8 \rangle.

∣v⃗∣=10|\vec{v}| = 10
√(36 + 64).
v⃗∣v⃗∣=⟨35,45⟩\dfrac{\vec{v}}{|\vec{v}|} = \left\langle \dfrac{3}{5}, \dfrac{4}{5} \right\rangle
6 ÷ 10: type the horizontal component.

Practice ★

  1. A vector has magnitude 12 and direction angle 120∘120^\circ. Find its exact components.Not to scale
  2. A vector has magnitude 3 and direction angle 120∘120^\circ. Find its exact components.Not to scale
  3. A vector has magnitude 3 and direction angle 60∘60^\circ. Find its exact components.Not to scale

Watch out!

Practice ★★

  1. Find the horizontal component of the unit vector in the direction of ⟨3,4⟩\langle 3, 4 \rangle.
  2. Find the horizontal component of the unit vector in the direction of ⟨15,8⟩\langle 15, 8 \rangle.
  1. A ball is kicked at 51 meters per second, at 35∘35^\circ above the ground. Find the horizontal component of its velocity, in meters per second, to 1 decimal place.Not to scale
  2. A ball is kicked at 40 meters per second, at 40∘40^\circ above the ground. Find the horizontal component of its velocity, in meters per second, to 1 decimal place.Not to scale
  3. A ball is kicked at 40 meters per second, at 70∘70^\circ above the ground. Find the horizontal component of its velocity, in meters per second, to 1 decimal place.Not to scale

Exit ticket

Three questions to check you have got it.

  1. A ball is kicked at 51 meters per second, at 35∘35^\circ above the ground. Find the horizontal component of its velocity, in meters per second, to 1 decimal place.Not to scale
  2. A ball is kicked at 40 meters per second, at 40∘40^\circ above the ground. Find the horizontal component of its velocity, in meters per second, to 1 decimal place.Not to scale
  3. A ball is kicked at 40 meters per second, at 70∘70^\circ above the ground. Find the horizontal component of its velocity, in meters per second, to 1 decimal place.Not to scale

Summary card

From magnitude and direction

  • ⟨rcos⁡θ, rsin⁡θ⟩\langle r\cos\theta,\ r\sin\theta \rangle.
  • Unit vector: v⃗∣v⃗∣\dfrac{\vec{v}}{|\vec{v}|}, length 1.

Watch out: horizontal goes with cosine, vertical with sine.

End of the lesson

Next lesson: The dot product and the angle between vectors Back to the chapter

Answers: Components from magnitude and direction; unit vectors

A

  1. 636 \sqrt{3}
  2. 332\frac{3 \sqrt{3}}{2}
  3. 332\frac{3 \sqrt{3}}{2}

B

  1. 35\frac{3}{5}
  2. 1517\frac{15}{17}

C

  1. 51cos⁡(7π36)51 \cos{\left(\frac{7 \pi}{36} \right)}
  2. 40cos⁡(2π9)40 \cos{\left(\frac{2 \pi}{9} \right)}
  3. 40cos⁡(7π18)40 \cos{\left(\frac{7 \pi}{18} \right)}

D

  1. 51cos⁡(7π36)51 \cos{\left(\frac{7 \pi}{36} \right)}
  2. 40cos⁡(2π9)40 \cos{\left(\frac{2 \pi}{9} \right)}
  3. 40cos⁡(7π18)40 \cos{\left(\frac{7 \pi}{18} \right)}

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