Question

Find each of the given projections if \(\mathbf{u}=3 \mathbf{i}+2 \mathbf{j}+\mathbf{k}, \mathbf{v}=2 \mathbf{i}-\mathbf{k}\), and \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}-3 \mathbf{k}\). \(\operatorname{proj}_{\mathbf{v}} \mathbf{u}\)

Short answer

\(\operatorname{proj}_{\mathbf{v}} \mathbf{u} = 2\mathbf{i} - \mathbf{k}\).

Step-by-step solution

Calculate Dot Product of u and v

First, we need to find the dot product of \(\mathbf{u}\) and \(\mathbf{v}\). The formula for the dot product is \(\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3\). For our vectors, \(\mathbf{u} = 3\mathbf{i} + 2\mathbf{j} + \mathbf{k}\) and \(\mathbf{v} = 2\mathbf{i} - \mathbf{k}\), the dot product is calculated as follows: \[\mathbf{u} \cdot \mathbf{v} = 3 \cdot 2 + 2 \cdot 0 + 1 \cdot (-1) = 6 - 1 = 5\]

Calculate Magnitude Squared of v

Next, compute the magnitude squared of \(\mathbf{v}\), denoted as \(|\mathbf{v}|^2\). The formula is \(|\mathbf{v}|^2 = v_1^2 + v_2^2 + v_3^2\). For \(\mathbf{v} = 2\mathbf{i} - \mathbf{k}\), we calculate: \[|\mathbf{v}|^2 = 2^2 + 0^2 + (-1)^2 = 4 + 0 + 1 = 5\]

Use Projection Formula

Now we use the projection formula to find \(\operatorname{proj}_{\mathbf{v}} \mathbf{u}\), which is given by: \[\operatorname{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|^2} \mathbf{v}\]Substitute the values we found: \[\operatorname{proj}_{\mathbf{v}} \mathbf{u} = \frac{5}{5} \cdot (2\mathbf{i} - \mathbf{k})\]Simplifying, we find: \[\operatorname{proj}_{\mathbf{v}} \mathbf{u} = 1 \cdot (2\mathbf{i} - \mathbf{k}) = 2\mathbf{i} - \mathbf{k}\]

Conclude the Projection

The projection of \(\mathbf{u}\) onto \(\mathbf{v}\) is thus \(\operatorname{proj}_{\mathbf{v}} \mathbf{u} = 2\mathbf{i} - \mathbf{k}\). This is the final result of our calculation, and it describes the component of \(\mathbf{u}\) that lies in the direction of \(\mathbf{v}\).

Key concepts

Dot Product

In vector mathematics, the dot product is a fundamental concept used to determine the relationship between two vectors. The dot product of two vectors, such as \( \mathbf{u} \) and \( \mathbf{v} \), is calculated using the formula \( \mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3 \). This operation effectively combines the components of each vector to produce a scalar, which is a single number.

For example, if \( \mathbf{u} = 3\mathbf{i} + 2\mathbf{j} + \mathbf{k} \) and \( \mathbf{v} = 2\mathbf{i} - \mathbf{k} \), their dot product would be calculated as follows:
\[ \mathbf{u} \cdot \mathbf{v} = 3 \times 2 + 2 \times 0 + 1 \times (-1) = 6 - 1 = 5 \]
This result, 5, tells us about the angle between the vectors; the closer it is to zero, the more perpendicular the vectors are. If it's positive, the vectors form an acute angle, and if negative, an obtuse angle.

Magnitude of a Vector

The magnitude of a vector, sometimes called the length or norm, is calculated to understand how long the vector is. The magnitude of a vector \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k} \) is given by \( |\mathbf{v}| = \sqrt{a^2 + b^2 + c^2} \).

However, in projection calculations, we often need the magnitude squared, \(|\mathbf{v}|^2\), because it simplifies the math. For our vector \( \mathbf{v} = 2\mathbf{i} - \mathbf{k} \), the magnitude squared is:
\[ |\mathbf{v}|^2 = 2^2 + 0^2 + (-1)^2 = 4 + 0 + 1 = 5 \]
This value helps in determining how much of one vector projects onto another.

Orthogonal Projection

Orthogonal projection involves capturing the shadow or the component of one vector along another vector's direction. This process is essential in numerous applications, like physics and computer graphics. The projection of vector \( \mathbf{u} \) onto \( \mathbf{v} \) is computed by first finding the dot product \( \mathbf{u} \cdot \mathbf{v} \), then dividing by the magnitude squared \(|\mathbf{v}|^2\) to find the scalar multiple of \( \mathbf{v} \).

This is represented mathematically as:
\[ \operatorname{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|^2} \mathbf{v} \]
In our example:
\[ \operatorname{proj}_{\mathbf{v}} \mathbf{u} = \frac{5}{5} (2\mathbf{i} - \mathbf{k}) = 2\mathbf{i} - \mathbf{k} \]
This projection describes the part of \( \mathbf{u} \) that aligns with \( \mathbf{v} \), effectively ignoring any component of \( \mathbf{u} \) that is perpendicular to \( \mathbf{v} \).

Vector Mathematics

Vector mathematics is a branch of mathematics that deals with quantities having both magnitude and direction. Vectors are often denoted as \( \mathbf{u}, \mathbf{v}, \text{ or } \mathbf{w} \) and can represent various physical quantities such as velocity, force, or displacement.

• Basic Operations: Include addition, subtraction, and finding dot and cross products.
• Applications: Used in engineering, physics, and computer science to analyze forces, movements, and other vector-related phenomena.
• Visualization: Often visualized as arrows in a coordinate system, where the arrow direction represents the vector direction and the length represents its magnitude.

Understanding these concepts through operations like projection allows us to solve real-world problems, including those related to mechanics and graphical computations. The combination of magnitude, direction, and operations provides a powerful toolset for dealing with two and three dimensional space.