Question

Vectors with equal projections Given a fixed vector \(\mathbf{v},\) there is an infinite set of vectors \(\mathbf{u}\) with the same value of \(\operatorname{proj}_{\mathbf{v}} \mathbf{u}\). Find another vector that has the same projection onto \(\mathbf{v}=\langle 1,1\rangle\) as \(\mathbf{u}=\langle 1,2\rangle .\) Draw a picture.

Short answer

Question: Find a vector that has the same projection onto \(\mathbf{v}=\langle 1,1\rangle\) as \(\mathbf{u}=\langle 1,2\rangle\). Answer: \(\mathbf{u'}=\left\langle \frac{5}{2}, \frac{1}{2} \right\rangle\).

Step-by-step solution

Find the projection of \(\mathbf{u}\) onto \(\mathbf{v}\)

The formula to find the projection of a vector \(\mathbf{u}\) onto a vector \(\mathbf{v}\) is given by: \(\operatorname{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}}\mathbf{v}\) Let's calculate the projection of \(\mathbf{u}=\langle 1,2\rangle\) onto \(\mathbf{v}=\langle 1,1\rangle\): \(\operatorname{proj}_{\mathbf{v}} \mathbf{u} = \frac{\langle 1, 2\rangle\cdot\langle 1,1\rangle}{\langle 1,1\rangle\cdot\langle 1,1\rangle} \langle 1,1\rangle = \frac{3}{2} \langle 1,1\rangle = \left\langle \frac{3}{2}, \frac{3}{2} \right\rangle\)

Vector with the same projection

Observe that any vector obtained by adding the projection vector to a vector orthogonal to \(\mathbf{v}\) will also have the same projection onto it. Let's find an orthogonal vector to \(\mathbf{v}\): A vector \(\mathbf{w}=\langle x, y\rangle\) is orthogonal to \(\mathbf{v}\) if their dot product is equal to 0, i.e. \(\mathbf{w}\cdot\mathbf{v}=0\). \(\langle x, y\rangle\cdot\langle 1,1\rangle = x+y=0\) Now, we can choose any value for \(x\), and obtain the corresponding value for \(y\) that satisfies the equation. For example, we can choose \(x=1\), then \(y=-1\), so \(\mathbf{w}=\langle 1, -1\rangle\) is orthogonal to \(\mathbf{v}\). Now, we can create a new vector, say \(\mathbf{u'}\), by adding the projection vector to the orthogonal vector we found. This new vector should have the same projection onto \(\mathbf{v}\) as \(\mathbf{u}\): \(\mathbf{u'}=\operatorname{proj}_{\mathbf{v}} \mathbf{u} + \mathbf{w} = \left\langle \frac{3}{2}, \frac{3}{2} \right\rangle + \left\langle 1, -1 \right\rangle = \left\langle \frac{5}{2}, \frac{1}{2} \right\rangle\)

Draw a picture

To draw a picture, plot \(\mathbf{v}=\langle 1,1\rangle\), \(\mathbf{u}=\langle 1,2\rangle\), and \(\mathbf{u'}=\left\langle \frac{5}{2}, \frac{1}{2} \right\rangle\) on a coordinate plane. Also, draw the projections of \(\mathbf{u}\) and \(\mathbf{u'}\) onto \(\mathbf{v}\). You should see that both projections are identical, and both the projections and the orthogonal vector form triangles similar to one another.

Key concepts

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and the linear transformations that operate on these spaces. It's a foundational topic for many fields including physics, engineering, computer science, and more. At the heart of linear algebra lies the concept of vectors, which are entities that possess both a magnitude and a direction. Think of them as arrows pointing from one point to another. These vectors can be added together and multiplied by scalars to create new vectors.

The operations and rules governing these vectors form the core principles of linear algebra. These operations include vector addition, scalar multiplication, and the dot product. Understanding these rules is essential for working with vectors in any context. For instance, in the exercise about vector projections, we apply linear algebra to determine another vector with an identical projection onto a given vector by exploiting the relationship between dot products and orthogonal vectors.

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in linear algebra used to determine the angle between two vectors and the length of projections. It's calculated by multiplying the corresponding components of two vectors and then summing the products. In mathematical notation, the dot product of two vectors \( \mathbf{u} \), \( \mathbf{v} \) is typically written as \( \mathbf{u} \cdot \mathbf{v} \).

When you calculate \( \mathbf{u} \cdot \mathbf{v} \), you're essentially measuring how much of one vector goes in the direction of another. This value plays a crucial role in finding the projection of one vector onto another, as seen in the exercise where the dot product is used to calculate \( \operatorname{proj}_{\mathbf{v}} \mathbf{u} \). The result of this operation gives us the magnitude of \( \mathbf{u} \) in the direction of \( \mathbf{v} \).

Orthogonal Vectors

Orthogonal vectors are two vectors that meet at a right angle (90 degrees), meaning they do not share any direction. In terms of the dot product, orthogonal vectors have a dot product of zero. This is because the cosine of 90 degrees is zero, and the dot product is equal to the product of the vectors' magnitudes multiplied by the cosine of the angle between them.

Understanding orthogonal vectors is essential when dealing with vector projections. If you add any vector that is orthogonal to your original vector \( \mathbf{v} \) to the projection of \( \mathbf{u} \) onto \( \mathbf{v} \), the resulting vector will have the same projection onto \( \mathbf{v} \) as \( \mathbf{u} \) itself. In the exercise, we use this property to find a new vector \( \mathbf{u'} \) that shares its projection with \( \mathbf{u} \), simply by finding a vector that's orthogonal to \( \mathbf{v} \) and adding it to the projection of \( \mathbf{u} \).