Projection

Definition

Projection measures the part of one vector that lies in the direction of another vector.

Simple image:

projection = shadow of one vector on another direction

Example:

a = [3, 4]
direction = [1, 0]

The part of a along [1, 0] is:

[3, 0]

Projection extracts the component of a vector along a direction.

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Projection Onto An Axis

Project:

a = [3, 4]

onto the first coordinate direction:

[1, 0]

The result is:

[3, 0]

Project onto the second coordinate direction:

[0, 1]

The result is:

[0, 4]

Projection can isolate how much a vector contributes to one direction.

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Projection Uses Alignment

Projection depends on how aligned the vectors are.

If vectors point in the same direction, projection is large.

Example:

a = [5, 0]
direction = [1, 0]
projection = [5, 0]

If they are perpendicular, projection is zero.

Example:

a = [0, 5]
direction = [1, 0]
projection = [0, 0]

Projection is based on directional alignment.

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Projection Length

Sometimes we only want the scalar amount in a direction.

If the direction vector has length 1, the scalar projection is:

dot(a, direction)

Example:

a = [3, 4]
direction = [1, 0]

dot(a, direction) = 3

So a has amount 3 in the [1, 0] direction.

Dot product with a unit direction gives projection amount.

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Projection Vector

Projection can also return a vector.

Example:

a = [3, 4]
direction = [1, 0]

Scalar amount:

3

Projection vector:

3 * [1, 0] = [3, 0]

Projection vector keeps the projected amount in the target direction.

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Projection Requires A Direction

You cannot project onto the zero vector as a direction.

Example:

direction = [0, 0]

This has no direction and length 0.

So it cannot define where the projection should land.

Projection needs a nonzero direction.

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Why Projection Matters For Data

Projection helps answer:

How much of this data vector lies in this direction?
How much does this feature direction explain?
What part remains outside this direction?

This matters because vector data often contains many components, but sometimes we want to study one direction at a time.

Projection breaks vector information into directional parts.

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Final Summary

• Projection measures the part of one vector along another direction.
• It is like a shadow on a direction.
• Projection depends on alignment.
• Dot product with a unit direction gives projection amount.
• Projection can return a scalar amount or a vector.
• Projection requires a nonzero target direction.