Dot Product

MIT 18.02 Multivariable Calculus, lecture 1, explained in plain English

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Part 1 · 0:00–10:00

This part explains what a vector is, how to read its parts, and how to find its length using the Pythagorean theorem.

Key ideas

A vector has both a direction and a length (called magnitude). You draw it as an arrow that points somewhere and has a set size. 1:09
A vector does not need a fixed starting point. If two arrows have the same length and point the same way, they are the same vector. 4:37
You can describe a vector by its components, which are how far it goes along the x, y, and z axes. For the vector 3, 2, 1, that means 3 units along x, 2 along y, and 1 along z. 6:30
To find the length, you use the Pythagorean theorem twice: first in the flat plane, then again with the up-and-down part added on. 8:00
For any vector with components a1, a2, a3, the length is the square root of a1 squared plus a2 squared plus a3 squared. This works even when there are more than three components. 9:16

Words to know

vector: A quantity that has both a direction and a length, shown as an arrow.
magnitude: The length or size of a vector, which is just a plain number.
scalar: An ordinary number with no direction, as opposed to a vector.
components: The separate amounts a vector moves along each coordinate axis, such as x, y, and z.

Example

Imagine walking 3 blocks east, then 2 blocks north, then 1 block up a ladder. Your walk is like the vector 3, 2, 1, and the straight-line distance from start to finish is its length. The lesson shows that this length is the square root of 14.

3 checked practice questions for this part are in the app.

Part 2 · 10:00–20:00

This part teaches how to add and scale vectors, and introduces the dot product, a way to multiply two vectors to get a single number.

Key ideas

Vectors can be understood two ways: as arrows you can draw, and as lists of numbers you can calculate with. Every question has a geometric answer and a numerical answer. 10:20
To add two vectors, you can put them head to tail and draw the diagonal of the parallelogram they form. Numerically, you just add the matching components. The order does not matter: A plus B equals B plus A. 11:09
Multiplying a vector by a scalar (a plain number) makes it longer or shorter, and a negative number also flips its direction. Subtracting a vector means adding its opposite. 13:06
The dot product multiplies two vectors by multiplying matching components and adding the results. The answer is a number, not a vector. 16:29
Geometrically, the dot product equals the length of A times the length of B times the cosine of the angle between them. This links the dot product to both lengths and angles. The lecturer says the proof comes next, starting with a vector dotted with itself. 17:42

Words to know

scalar: A plain number, with no direction. It is used to stretch or shrink a vector.
component: One of the numbers in a vector's list, such as the first, second, or third number, that describes how far it goes along each axis.
dot product: A way to multiply two vectors that gives one number. You multiply matching components and add them up.
cosine: A number that depends on an angle. It is used here to connect the angle between two vectors to the dot product.

Example

Imagine walking 3 blocks east and then 2 blocks north. You end up in the same place as walking 2 blocks north first and then 3 blocks east. That is why adding vectors in either order gives the same result.

3 checked practice questions for this part are in the app.

Part 3 · 20:00–30:00

This part shows how the dot product connects to the law of cosines, and how it can be used to find the angle between two lines in space.

Key ideas

The dot product of a vector with itself gives the square of its length. This was checked earlier, and it is the starting point here. 20:00
The law of cosines is a rule about the sides of any triangle. It adds a third term, 2 times the two side lengths times the cosine of the angle between them, to the Pythagoras rule. 21:00
If you expand the dot product of (A minus B) with itself, you get A dot A, minus A dot B, minus B dot A, plus B dot B. Since A dot B and B dot A are equal, this matches the law of cosines, which shows that A dot B equals length A times length B times the cosine of the angle. 23:49
To find an angle, you can use the dot product formula. Find the dot product and the two lengths, then divide the dot product by the lengths to get the cosine of the angle. 28:31
To find the vector from P to Q, move -1 along x, 1 along y, and 0 along z. The lesson then starts on the vector from P to R, but it does not finish that step in this part. 29:27

Words to know

dot product: A way to multiply two vectors that gives a single number. It can be used to find lengths and angles.
law of cosines: A rule that links the three side lengths of a triangle to the angle between two of its sides.
cosine: A number tied to an angle. It helps connect an angle to side lengths in a triangle.
vector: A list of numbers that describes a movement or direction, such as moving 1 step along x and 2 steps along z.

Example

Imagine two friends walking from the same corner, each on a different road. The angle between their paths and the distances they walked form a triangle, just like the vectors A and B in the lesson. If you know how far each one walked and the dot product of their paths, you can work out the angle between them without measuring it on a map.

3 checked practice questions for this part are in the app.

Part 4 · 30:00–38:44

This part shows how the dot product tells you whether two vectors point the same way, point opposite ways, or are perpendicular, and how it can describe a plane.

Key ideas

To get a dot product, multiply the matching parts of two vectors and add the results. Dividing by the lengths of the vectors gives the cosine of the angle between them. 30:40
The sign of the dot product matches the sign of the cosine. If the angle is less than 90 degrees, the dot product is positive, so the vectors mostly go the same way. 31:48
If the angle is exactly 90 degrees, the dot product is zero. If the angle is more than 90 degrees, the dot product is negative, so the vectors go in opposite directions. 32:33
The equation x + 2y + 3z = 0 describes a plane. Each point on it is one where the vector from the origin to that point is perpendicular to the vector (1, 2, 3). 35:32
A dot product of zero is a quick test for perpendicular vectors. You do not need to measure the angle to check. 37:25

Words to know

dot product: A way to combine two vectors into one number. You multiply matching parts and add them up.
cosine: A number that describes an angle. Here it tells you how much two vectors line up with each other.
orthogonality: A fancy word that means perpendicular, or meeting at a right angle (90 degrees).
perpendicular: Meeting at a right angle, like the corner of a square or a wall meeting the floor.

Example

Think of a flat table and a pencil standing straight up from its surface. The pencil is perpendicular to the table, so the angle between them is 90 degrees. If you lean the pencil over a little, it is still pointing somewhat upward, and the angle is less than 90 degrees.

3 checked practice questions for this part are in the app.

Lecture 8: Partial Derivatives →

Notes written by AI (Claude) from the lecture's official transcript, so check anything important against the lecture. MIT OpenCourseWare · CC BY-NC-SA 4.0 · not affiliated with or endorsed by MIT. These notes are shared under the same license.

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