Question

Given \(\mathbf{v}=\left[\begin{array}{l}x \\ y \\ z\end{array}\right]\) in component form, show that the projections of \(\mathbf{v}\) on \(\mathbf{i}, \mathbf{j},\) and \(\mathbf{k}\) are \(x \mathbf{i}, y \mathbf{j},\) and zk, respectively.

Short answer

Projections are \(x \mathbf{i}, y \mathbf{j}, z \mathbf{k}\).

Step-by-step solution

Understand Vector Components

The vector \(\mathbf{v}\) is given in component form as \(\mathbf{v} = \begin{bmatrix} x \ y \ z \end{bmatrix}\), where \(x\), \(y\), and \(z\) are the scalar components of the vector along the standard unit vectors \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\), respectively.

Projections on Standard Unit Vectors

To find the projection of \(\mathbf{v}\) on \(\mathbf{i}\), we consider the vector as \(x \mathbf{i}\). Similarly, for projection on \(\mathbf{j}\), it is \(y \mathbf{j}\), and for \(\mathbf{k}\), it is \(z \mathbf{k}\).

Formula for Vector Projection

The projection of a vector \(\mathbf{a}\) on a vector \(\mathbf{b}\) is given by the formula: \[ \text{proj}_{\mathbf{b}} \mathbf{a} = \left( \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \right) \mathbf{b} \].

Apply Projection Formula

Using the formula, calculate the projections:- For \(\mathbf{i}\): \(\text{proj}_{\mathbf{i}} \mathbf{v} = \left( \frac{x \cdot 1}{1} \right) \mathbf{i} = x \mathbf{i}\).- For \(\mathbf{j}\): \(\text{proj}_{\mathbf{j}} \mathbf{v} = \left( \frac{y \cdot 1}{1} \right) \mathbf{j} = y \mathbf{j}\).- For \(\mathbf{k}\): \(\text{proj}_{\mathbf{k}} \mathbf{v} = \left( \frac{z \cdot 1}{1} \right) \mathbf{k} = z \mathbf{k}\).

Conclusion

Thus, the projections of \(\mathbf{v} = \begin{bmatrix} x \ y \ z \end{bmatrix}\) on \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) are respectively \(x \mathbf{i}\), \(y \mathbf{j}\), and \(z \mathbf{k}\). These represent the vector \(\mathbf{v}\) split into its scalar components along each axis.

Key concepts

Vector Components

Vector components are the building blocks of a vector. Each vector is composed of individual scalar quantities aligned along specific directions. Let's consider vector \( \mathbf{v} \) expressed as \( \begin{bmatrix} x \ y \ z \end{bmatrix} \). Here, each component, \(x\), \(y\), and \(z\), represents a direction along orthogonal axes.

These axes are usually the coordinate plane's standard unit vectors \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \), corresponding to the x, y, and z axes. The components tell us how far and in which direction the vector extends along each axis.

- **\(x\)**: The scalar component along the \(x\)-axis, indicating the vector's horizontal distance.
- **\(y\)**: The scalar component along the \(y\)-axis, indicating the vector's vertical distance.
- **\(z\)**: The scalar component along the \(z\)-axis, indicating the vector's depth in three dimensions.

Recognizing vector components provides a deeper understanding of how vectors behave in different spaces and are fundamental in physics and engineering.

Unit Vectors

Unit vectors are vectors with a magnitude of one, used to specify directions in vector space without providing magnitude. They are fundamental in vector algebra to express vector direction.

- **Standard Unit Vectors**:
- **\(\mathbf{i}\)**: Points along the x-axis.
- **\(\mathbf{j}\)**: Points along the y-axis.
- **\(\mathbf{k}\)**: Points along the z-axis.

Since they have a magnitude of one, when a vector like \( \mathbf{v} = \begin{bmatrix} x \ y \ z \end{bmatrix} \) is projected along these unit vectors, each scalar directly scales these unit vectors. For instance, \(x\mathbf{i}\) shows that the vector \( \mathbf{v} \) extends \(x\) units in the x direction. Understanding unit vectors simplifies many calculations in vector algebra, ensuring clear representation of directions.

Projection Formula

The projection formula is crucial in determining how one vector's direction and length are mapped onto another vector. This concept is vital for resolving forces, velocities, and other vector quantities in different directions.

To find the projection of one vector \( \mathbf{a} \) onto another vector \( \mathbf{b} \), use the formula:
\[ \text{proj}_{\mathbf{b}} \mathbf{a} = \left( \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \right) \mathbf{b} \]
This formula calculates the scalar that scales \( \mathbf{b} \) so that it reflects \( \mathbf{a} \)’s influence in \( \mathbf{b} \)’s direction.

- **Where**:
- **\( \mathbf{a} \cdot \mathbf{b} \)**: The dot product providing directional compatibility between \( \mathbf{a} \) and \( \mathbf{b} \).
- **\( \mathbf{b} \cdot \mathbf{b} \)**: The dot product of \( \mathbf{b} \) with itself, equating to \( \|\mathbf{b}\|^2 \).

In practical terms, when \( \mathbf{v} \) is projected onto the standard unit vectors (such as \( \mathbf{i} \)), each projection shows how much of \( \mathbf{v} \) aligns with \( \mathbf{i} \). Thus, the projections \( \text{proj}_{\mathbf{i}} \mathbf{v} = x \mathbf{i} \) demonstrate \( \mathbf{v} \)'s alignment with the x-axis, and similarly for the others.