# How to Solve for Properties and Proofs of the Projection of Vectors for Calculus

Sometimes as we analyze vectors we are interested in finding out the individual components of a vector and where they are in relation to another but different vector. We may need to know how much each part points away from another vector and how much is along-side it. When we look at how much of a vector component is parallel to another vector, we are looking at the *projection* of that vector. Take a look at the following setup.

If I have two non-zero vectors **a **and **b**, I can break **a** down into the components **c**_{1} and **c**_{2} so that **a** = **c**_{1} + **c**_{2}. These components are special because **c**_{1} is parallel to **b **and **c**_{2} is orthogonal to **b**. Because **c**_{1} lies on **b** we say that it is the component of **a** onto **b**, or that the projection of **a** onto **b ** is **c**_{1}. We notate this as

**c**_{1 } = proj_{a}b

And don’t forget, **c**_{2} is orthogonal to **b**. It is also important to point out something you may observe as you use the projection. If the angle between the vectors is acute, then the projection points in the same direction as **b **and will seem to overlap portions of **b**, but if the angle between vectors is obtuse then the projection will point in the opposite direction as **b** and will not overlap it. The projection will still be parallel to **b** but the direction it points in depends on that angle (Larson 785).

## Using the Dot Product to Find Projections

Now, it is great if you have the above information (knowing **a, b, c**_{1}, and **c**_{2}) because finding the projection is simply just extracting the correct information and inserting it into the proper notation. But what if you only knew **a **and **b**? How could you find that projection of **a** onto **b**? Well, note that **c**_{1} has the length of c_{1} (since it is just a component of **a**, the magnitude is just ||**c**_{1}|| = [(c_{1})^{2}]^{0.5} = c_{1}) and it is in the direction of **b**. But to find a vector in that direction, I need to know the unit vector along that particular vector, in this case **b**. The unit vector in the direction of **b** would be **b** / ||**b**||. So,

**c**_{1} = c_{1}(**b/ ||b||)**

But what does the c_{1} equal? As previously mentioned, c_{1} is a component of **a**. We need to use the angle between the vectors and basic trigonometry to find the component that is along **b**. Specifically,

c_{1} = ||**a**||CosΘ

But CosΘ = (**a ∙ b) / **(||**a**||||**b**||), so

c_{1} = ||**a**|| (**a ∙ b) / **(||**a**||||**b**||) = (**a ∙ b) / **||**b**||

So, the projection of **a** onto **b **is

**c**_{1} = [(**a ∙ b) / **||**b**||] (**b/ ||b||) = b **(**a ∙ b) / (||b||**^{2}) (786)

## The Scalar Version of Projections

So now that we know what the projection of **a** onto **b **is, it may be insightful to observe the following:

**c**_{1} = **b **(**a ∙ b) / (||b||**^{2}) = [(**a ∙ b)/||b||] **(**b/||b||**) = k (**b/||b||**)

We can call k = (**a ∙ b)/||b||** the component of **a** in the direction of **b** because the **b/||b||** term is just a unit vector in the direction of **b **and k simply tells us the scalar amount of **a** that will end up being in that direction. It helps us see the underlining meaning behind many unusual terms and lots of jargon (786).

## Works Cited

Larson, Ron, Robert Hostetler, and Bruce H. Edwards. Calculus: Early Transcendental Functions. Maidenhead: McGraw-Hill Education, 2007. Print. 785-6.

- Basic Vector Properties and Proofs

Vectors are an important component of higher-level calculus. How do we arrive at many of the results that we take for granted? - Unit Vector Properties and Proofs

Unit vectors tell us the basic information of a vector. It is critical to know all the properties and proofs that help guide what we can do with them.

**© 2014 Leonard Kelley**

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