Question

Calculating orthogonal projections For the given vectors \(\mathbf{u}\) and \(\mathbf{v},\) calculate proj\(_{\mathbf{v}} \mathbf{u}\) and \(\operatorname{scal}_{\mathbf{v}} \mathbf{u}\). $$\mathbf{u}=\mathbf{i}+4 \mathbf{j}+7 \mathbf{k} \text { and } \mathbf{v}=2 \mathbf{i}-4 \mathbf{j}+2 \mathbf{k}$$

Short answer

Question: Calculate the orthogonal projection of vector \(\mathbf{u} = (1, 4, 7)\) onto vector \(\mathbf{v} = (2, -4, 2)\) and find the scaling factor of vector \(\mathbf{u}\) related to vector \(\mathbf{v}\). Answer: Step 1: Calculate the dot product between \(\mathbf{u}\) and \(\mathbf{v}\): \(\mathbf{u} \cdot \mathbf{v} = (1)(2) + (4)(-4) + (7)(2) = 2 - 16 + 14 = 0\) Step 2: Calculate the squared norm of \(\mathbf{v}\): \(\|\mathbf{v}\|^2 = (2)^2 + (-4)^2 + (2)^2 = 4 + 16 + 4 = 24\) Step 3: Calculate scaling factor \(\operatorname{scal}_{\mathbf{v}} \mathbf{u}\): \(\operatorname{scal}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2} = \frac{0}{24} = 0\) Step 4: Calculate proj\(_{\mathbf{v}} \mathbf{u}\): \(\operatorname{proj}_{\mathbf{v}} \mathbf{u} = \operatorname{scal}_{\mathbf{v}} \mathbf{u} \cdot \mathbf{v} = 0 \cdot (2, -4, 2) = (0, 0, 0)\) The orthogonal projection of vector \(\mathbf{u}\) onto vector \(\mathbf{v}\) is \((0, 0, 0)\), and the scaling factor is \(0\).

Step-by-step solution

Calculate the dot product between \(\mathbf{u}\) and \(\mathbf{v}\)

To calculate the dot product between \(\mathbf{u}\) and \(\mathbf{v}\), multiply corresponding components of each vector and then sum the products: $$\mathbf{u} \cdot \mathbf{v} = (1)(2) + (4)(-4) + (7)(2)$$

Calculate the squared norm of \(\mathbf{v}\)

To calculate the squared norm of \(\mathbf{v}\), square each component of \(\mathbf{v}\) and sum the squares: $$\|\mathbf{v}\|^2 = (2)^2 + (-4)^2 + (2)^2$$

Calculate scaling factor \(\operatorname{scal}_{\mathbf{v}} \mathbf{u}\)

Using the calculated dot product and squared norm, compute the scaling factor: $$\operatorname{scal}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2}$$

Calculate proj\(_{\mathbf{v}} \mathbf{u}\)

Finally, to find the orthogonal projection, multiply the obtained scaling factor with vector \(\mathbf{v}\): $$\operatorname{proj}_{\mathbf{v}} \mathbf{u} = \operatorname{scal}_{\mathbf{v}} \mathbf{u} \cdot \mathbf{v}$$

Key concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in vector algebra that expresses the product of two vectors. It is calculated by multiplying the corresponding components of two vectors and then summing the results. In mathematical terms, for two vectors \(\mathbf{a}\) and \(\mathbf{b}\), their dot product \(\mathbf{a} \cdot \mathbf{b}\) is defined as: \[\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 + \ldots + a_nb_n\] where \(a_i\) and \(b_i\) are the components of the vectors \(\mathbf{a}\) and \(\mathbf{b}\) in the \(i^{th}\) dimension.

The dot product has important implications in geometry: when two vectors are perpendicular, their dot product is zero. This property is essential when calculating the orthogonal projection, as it helps to determine if a vector is directly in line with another or if it lies in a completely different direction.

Vector Norm

The vector norm, often simply called the 'norm', measures the length or the magnitude of a vector. It is denoted as \(\|\mathbf{v}\|\) for a vector \(\mathbf{v}\). The norm can be calculated by taking the square root of the dot product of the vector with itself, which is equivalent to squaring each component of the vector, summing those squares, and then taking the square root. In more technical terms:

\[\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2 + \ldots + v_n^2}\]
This computation is essential in the normalization of vectors (creating a vector of length 1), determining distances between points (in space or on planes), and, as discussed here, calculating orthogonal projections and scaling factors. The square of the vector norm, \(\|\mathbf{v}\|^2\), directly contributes to computing these projections and scaling factors without the need for taking a square root.

Scaling Factor

The scaling factor, in the context of vector projections, refers to the scalar value by which a vector is multiplied to achieve the projection onto another vector. It represents how much of one vector is in the direction of another.

When you project vector \(\mathbf{u}\) onto vector \(\mathbf{v}\), as in the exercise, the scaling factor \(\operatorname{scal}_{\mathbf{v}} \mathbf{u}\) scales vector \(\mathbf{v}\) such that its endpoint lies on the line extended by \(\mathbf{v}\) and is closest to the endpoint of \(\mathbf{u}\). Mathematically, Step 3 in the given solution expresses the scaling factor as a ratio of the dot product \(\mathbf{u} \cdot \mathbf{v}\) to the squared norm of \(\|\mathbf{v}\|^2\). This ratio is crucial because it signifies how \(\mathbf{u}\) is stretched or shrunk in the direction of vector \(\mathbf{v}\).

Vector Algebra

Vector algebra is a branch of mathematics that deals with operations on vectors, which are quantities having both magnitude and direction. It includes operations like addition, subtraction, and scalar multiplication. Orthogonal projections are one of the many applications of vector algebra.

Understanding the algebraic properties of vectors allows us to perform computations that describe physical phenomena, computer graphics, and more. For instance, adding vectors represents combining forces in physics, while in navigation, it can represent combining motion vectors to understand the resulting path.

Vector algebra is inherently related to the geometric interpretations of vectors, which is why it serves as the backbone for calculating projections and understanding their behavior in multidimensional space. A solid grasp of vector algebra is essential for anyone endeavoring into fields such as engineering, physics, computer science, or any domain that relies on the mathematical representation of space and movement.