Length and the dot product
Slide 1 built vectors by stretching and adding. Now we measure them. The dot product takes two vectors and gives back one number: multiply the matching parts, then add. For 21 and 34 that is 2·3 + 1·4 = 10. This one number answers two everyday questions: how long is an arrow, and do two arrows meet at a right angle? It is also the small calculation inside multiplying a matrix by a vector (a matrix is a table of numbers; that is the next slide).
Lesson 1 · slide 2Dot product- v · w: multiply matching parts and add. The answer is a number, not a vector.
- Length: ‖v‖ = √(v · v). For 34 it is √25 = 5.
- v · w = 0 means v and w meet at a right angle (they are perpendicular).
- A unit vector has length 1. Divide any vector by its length to get one.
Multiply matching parts, then add
Line the two vectors up. Multiply the first parts, multiply the second parts, and add the results:
The order doesn't matter: w · v is also 10. Vectors with more parts work the same way, with more products to add:
Length
Dot a vector with itself. For v = 34, v · v = 3·3 + 4·4 = 25. That is the length squared. Why? The arrow is the long side of a right triangle whose other sides are 3 and 4, and Pythagoras says 3² + 4² = 5². So the length is
In 3D, add one more square: ‖122‖ = √(1 + 4 + 4) = √9 = 3. A vector with length 1 is called a unit vector. To make one, divide v by its length: 34 / 5 = 0.60.8 (dividing by 5 means multiplying by 1/5).
Zero means a right angle
21 · −12 = −2 + 2 = 0. Draw the two arrows and they meet at a right angle. That is always true: v · w = 0 exactly when v and w are perpendicular.
The sign tells you more. Positive: the arrows point roughly the same way (angle under 90°). Negative: they point roughly opposite ways (angle over 90°). Strang turns this into an exact rule for the angle, v · w = ‖v‖ ‖w‖ cos θ (θ is the angle between the arrows), which lesson 5 uses.
A bigger example: a shopping bill
Bread costs 2 euros, milk 1 and apples 3. You buy 1 bread, 4 milk and 2 apples. Prices p = 213 and amounts q = 142. The bill is one dot product:
A shop with 5,000 products does exactly the same thing, with 5,000 products to add.
Next: A matrix is a table of numbers (next slide). Each row of a matrix, dotted with a vector, gives one number. Do that for every row and you have multiplied a matrix by a vector.
Examples
Example 1 · A negative dot product
Find 1−2 · 34. What does the sign say?
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Negative, so the angle between the arrows is more than 90°.
Example 2 · Length and unit vector
Find the length of v = 68 and the unit vector in its direction.
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Example 3 · Make a right angle
Find c so that 1c is perpendicular to 42.
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So 1−2 meets 42 at a right angle.
Example 4
Are 123 and 30−1 perpendicular?
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1·3 + 2·0 + 3·(−1) = 3 + 0 − 3 = 0. Yes, they meet at a right angle.
Example 5
Show that 34 and 4−3 are perpendicular and have the same length.
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34 · 4−3 = 12 − 12 = 0, so they are perpendicular. Both lengths are √(9 + 16) = 5.
Questions and comments
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