Vector projection

Key takeaways

Vector projection belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Vector projection to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Vector projection from memory before moving on to harder problems.

Reference excerpt

The vector projection (also known as the vector component or vector resolution) of a vector a on (or onto) a non-zero vector b is the orthogonal projection of a onto a straight line parallel to b. The projection of a onto b is often written as proj b ⁡ a {\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} } or a∥b. The vector component or vector resolute of a perpendicular to b, sometimes also called the vector rejection of a from b (denoted oproj b ⁡ a {\displaystyle \operatorname {oproj} _{\mathbf {b} }\mathbf {a} } or a⊥b), is the orthogonal projection of a onto the plane (or, in general, hyperplane) that is orthogonal to b. Since both proj b ⁡ a {\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} } and oproj b ⁡ a {\displaystyle \operatorname {oproj} _{\mathbf {b} }\mathbf {a} } are vectors, and their sum is equal to a, the rejection of a from b is given by: oproj b ⁡ a = a − proj b ⁡ a . {\displaystyle \operatorname {oproj} _{\mathbf {b} }\mathbf {a} =\mathbf {a} -\operatorname {proj} _{\mathbf {b} }\mathbf {a} .}

To simplify notation, this article defines a 1 := proj b ⁡ a {\displaystyle \mathbf {a} _{1}:=\operatorname {proj} _{\mathbf {b} }\mathbf {a} } and a 2 := oproj b ⁡ a . {\displaystyle \mathbf {a} _{2}:=\operatorname {oproj} _{\mathbf {b} }\mathbf {a} .}

Thus, the vector a 1 {\displaystyle \mathbf {a} _{1}} is parallel to b , {\displaystyle \mathbf {b} ,} the vector a 2 {\displaystyle \mathbf {a} _{2}} is orthogonal to b , {\displaystyle \mathbf {b} ,} and a = a 1 + a 2 . {\displaystyle \mathbf {a} =\mathbf {a} _{1}+\mathbf {a} _{2}.} The projection of a onto b can be decomposed into a direction and a scalar magnitude by writing it as a 1 = a 1 b ^ {\displaystyle \mathbf {a} _{1}=a_{1}\mathbf {\hat {b}} }

where a 1 {\displaystyle a_{1}} is a scalar, called the scalar projection of a onto b, and b̂ is the unit vector in the direction of b. The scalar projection is defined as

a 1 = ‖ a ‖ cos ⁡ θ = a ⋅ b ^ {\displaystyle a_{1}=\left\|\mathbf {a} \right\|\cos \theta =\mathbf {a} \cdot \mathbf {\hat {b}} }

where the operator ⋅ denotes a dot product, ‖a‖ is the length of a, and θ is the angle between a and b. The scalar projection is equal in absolute value to the length of the vector projection, with a minus sign if the direction of the projection is opposite to the direction of b, that is, if the angle between the vectors is more than 90 degrees. The vector projection can be calculated using the dot product of a {\displaystyle \mathbf {a} } and b {\displaystyle \mathbf {b} } as:

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Vector projection

Start with the simplest possible case. Write down what Vector projection claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Vector projection before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Vector projection ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Vector projection

In research: Vector projection appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Vector projection in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Vector projection is common in secondary-school and first-year university syllabi. It links to neighbouring topics Functions and mappings, Operations on vectors, Transformation (function), so understanding it makes those chapters shorter.
In everyday life: Look for Vector projection outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Vector projection in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Vector projection means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Vector projection out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Vector projection in simple terms?

The vector projection (also known as the vector component or vector resolution) of a vector a on (or onto) a non-zero vector b is the orthogonal projection of a onto a straight line parallel to b. The projection of a onto b is often written as proj b ⁡ a {\displaystyle \operatorname {proj} _{\mathb…

Why does Vector projection matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Vector projection?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Vector projection.