Topic 5 of 8

Vectors

Step 14 · Trigonometry and precalculus topics

Describe vectors by components and by magnitude and direction, add and scale them, and use the dot product to find angles and test for perpendicular vectors.

Components and magnitude

A vector has a size (its magnitude) and a direction, like a velocity or a force. In components, ⟨3, 4⟩ means 3 across and 4 up.

The vector ⟨3, 4⟩: 3 across and 4 up, with magnitude 5.

|\langle a, b\rangle| = \sqrt{a^{2} + b^{2}}

Example: From two points

Find the vector from P(2, −1) to Q(7, 11), and its magnitude.

Across: 7 − 2 = 5. Up: 11 − (−1) is 12. So ⟨5, 12⟩.
√(25 + 144) = 13.

Answer⟨5, 12⟩, with magnitude 13

Example: From size and direction

A force of 10 N acts at 30° above the horizontal. Find its components.

Across: 10 cos 30°, about 8.66.
Up: 10 sin 30° = 5.

AnswerAbout ⟨8.66, 5⟩

Adding and scaling vectors

Add vectors by adding their components; in a picture, place them head to tail. Multiplying by a number scales the length, and a negative number reverses the direction.

Example: Adding and scaling

Find ⟨3, −2⟩ + ⟨−1, 6⟩ and 2⟨3, −2⟩.

Across: 3 + (−1) is 2. Up: −2 + 6 = 4.
Doubling each part gives ⟨6, −4⟩.

Answer⟨2, 4⟩ and ⟨6, −4⟩

Head to tail: ⟨4, 1⟩ followed by ⟨1, 3⟩ ends at the same point as ⟨5, 4⟩.

Example: A unit vector

Find the unit vector, of length 1, pointing the same way as ⟨6, 8⟩.

Its magnitude is 10.
Divide each component by 10.

Answer⟨0.6, 0.8⟩

Example: A boat and a current

A boat heads north at 8 km/h across a current flowing east at 3 km/h. How fast does it actually move?

Its velocity is ⟨3, 8⟩.
√(9 + 64) = √73, about 8.54.

AnswerAbout 8.54 km/h

The dot product

The dot product multiplies matching components and adds: ⟨a, b⟩ · ⟨c, d⟩ = ac + bd. It is a number, and it measures how much two vectors point the same way.

\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos\theta

Example: A dot product

Find ⟨2, 5⟩ · ⟨3, −1⟩.

2 × 3 + 5 × (−1) is 6 − 5 = 1.

Answer1

Example: Perpendicular vectors

Are ⟨4, 6⟩ and ⟨3, −2⟩ perpendicular?

4 × 3 + 6 × (−2) is 12 − 12 = 0.

AnswerYes: a dot product of 0 means a right angle

Example: The angle between

Find the angle between ⟨1, 0⟩ and ⟨1, 1⟩.

The dot product is 1, and the magnitudes are 1 and √2: $\cos\theta = \frac{1}{\sqrt{2}}$ .
θ = 45°.

Answer45°

Try it yourself

Work each one on paper first, then open the answer to check.

Find the magnitude of ⟨−7, 24⟩.
Show answer
25
Find the vector from (1, 3) to (4, −1).
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⟨3, −4⟩
Find 3⟨2, −1⟩ − ⟨4, 5⟩.
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⟨2, −8⟩
Find ⟨1, 2⟩ · ⟨3, 4⟩.
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11
Are ⟨2, 3⟩ and ⟨−3, 2⟩ perpendicular?
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Yes: −6 + 6 = 0.
A vector has magnitude 20 and direction 60°. Find its components.
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⟨10, 10√3⟩, about ⟨10, 17.3⟩