Scalar Projection & Vector Projection

Refer to the note in Pre Linear algebra about understanding Dot product.

Assume that the vector w projects onto the vector v.
Notation:

• Scalar projection: Componentᵥw, read as "Component of w onto v".
• Vector projection: Projectionᵥw, read as "Projection of w onto v".

Notice that: When you read it, it’s in a reverse order! Very important!

Projection Formula

Note that, the formula concerns of these concepts as prerequisites:

• Dot product calculation
• Dot product cosine formula
• Unit vector

How to calculate the Scalar Projection

The name is just the same with the names mentioned above: boosting.

Componentᵥw = (dot product of v & w) / (w's length)

Refer to lecture by Imperial College London: Projection
Refer also to Khan academy: Intro to Projections

What if we know the vectors, and we want to know how much is the Scalar projection(the shadow)?
Example:

How we’re gonna solve this is: We know the vectors, so we can get their dot producteasily by taking their linear combination; and we know the length of each vector, by using Pythagorean theorem; and then we get the projection, as in the picture.

How to calculate the Vector Projection

It’s another idea for projection, and less intuitive.

Remember that a Scalar projection is the vector's LENGTH projected on another vector. And when we add the DIRECTION onto the LENGTH, it became a vector, which lies on another vector. Then it makes it a Vector projection.

It can be understood as this formula:

Projectionᵥw = (Componentᵥw) * (Unit vector of v)

But usually we write it as this:

Refer also to video for formula by Kate Penner: Vector Projection Equations
Refer to video by Firefly Lectures: Vector Projections — Example 1

Example:

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