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## Linear algebra

### Course: Linear algebra>Unit 2

Lesson 2: Linear transformation examples

# Introduction to projections

Determining the projection of a vector on s line. Created by Sal Khan.

## Want to join the conversation?

• Explain projection of a vector •  The shadow is the projection of your arm (one vector) relative to the rays of the sun (a second vector).
• Just a quick question, at you cannot cancel the top vector v and the bottom vector v right? Is this because they are dot products and not multiplication signs? • Please remind me why we CAN'T reduce the term (x*v / v*v) to (x / v), like we could if these were just scalars in numerator and denominator... but we CAN distribute ((x - c*v) * v) to get (x*v - c*v*v) ?
Where do I find these "properties" (is that the correct word? i.e. what I can and can't transform in a formula), preferably all conveniently** listed?
Presumably, coming to each area of maths (vectors, trig functions) and not being a mathematician, I should acquaint myself with some "rules of engagement" board (because if math is like programming, as Stephen Wolfram said, then to me it's like each area of maths has its own "overloaded" -, +, * operators. But where is the doc file where I can look up the "definitions"??).
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**i.e. without diving into Ancient Greek or Renaissance history ;)_ • At , how can you multiply vectors such a way? They are (2x1) and (2x1). Can they multiplied to each other in a first place? • What does orthogonal mean? • hi there, how does unit vector differ from complex unit vector? • Many vector spaces have a norm which we can use to tell how large vectors are. R^2 has a norm found by ||(a,b)||=a^2+b^2. The complex vectors space C also has a norm given by ||a+bi||=a^2+b^2. The look similar and they are similar. Unit vectors are those vectors that have a norm of 1. There is a pretty natural transformation from C to R^2 and vice versa so you might think of them as the same vector space. But they are technically different and if you get more advanced with what you are doing with them (like defining a multiplication operation between vectors) that you want to keep them distinguished.
• Why not mention the unit vector in this explanation? • v actually is not the unit vector. The unit vector for L would be (2/sqrt(5), 1/sqrt(5)) .

If you want to solve for this using unit vectors here's an alternative method that relates the problem to the dot product of x and v in a slightly different way:

First, the magnitude of the projection will just be ||x||cos(theta), the dot product gives us x dot v = ||x||*||v||*cos(theta) , therefore ||x||*cos(theta) = (x dot v) / ||v|| . This gives us the magnitude so if we now just multiply it by the unit vector of L this gives our projection (x dot v) / ||v|| * (2/sqrt(5), 1/sqrt(5)) . Which is equivalent to Sal's answer.
• Since dot products "means" the "same-direction-ness" of two vectors (ie. if the two vectors are perpendicular, the dot product is 0; as the angle between them get smaller and smaller, the dot product gets bigger). According to the equation Sal derived, the scaling factor is ("same-direction-ness" of vector x and vector v) / (square of the magnitude of vector v). Does it have any geometrical meaning? How does it geometrically relate to the idea of projection?

Thank you in advance! I hope I could express my idea more clearly... • The factor 1/||v||^2 isn't thrown in just for good luck; it's based on the fact that unit vectors are very nice to deal with. If you're in a nice scalar field (such as the reals or complexes) then you can always find a way to "normalize" (i.e. make the length 1) of any vector. The dot product is exactly what you said, it is the projection of one vector onto the other. You can draw a nice picture for yourself in R^2 - however sometimes things get more complicated. Later on, the dot product gets generalized to the "inner product" and there geometric meaning can be hard to come by, such as in Quantum Mechanics where up can be orthogonal to down.  