Problem 58 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the projection of \(\mathbf{u}\) onto \(\mathbf{v}\). Then write \(\mathbf{u}\) as the sum of two orthogonal vectors, one of which is \({\mathrm{proj}_v}\mathrm{u}.\) $$\begin{aligned} &\mathbf{u}=\langle 4,2\rangle\\\ &\mathbf{v}=\langle 1,-2\rangle \end{aligned}$$

Step-by-Step Solution

Answer

The projection of u onto v is \(\langle -0.8, 1.6 \rangle\) and u can be expressed as the sum of this projection and the orthogonal vector \(\langle 4.8, 0.4 \rangle\).

Step 1: Calculate Scalar Projection

First calculate the scalar projection of u onto v. We can find it using the formula \(c = {u \cdot v \over ||v||^2}\). Substituting u and v by their values (from exercise), we get \(c = {\langle 4,2\rangle \cdot \langle 1,-2\rangle \over ||\langle 1,-2\rangle||^2}\). Computing it, we obtain c = -0.8.

Step 2: Calculate Vector Projection

Next step is to find the vector projection. The formula to find vector projection is \(proj_v u= c \cdot v\). Substituting c and v by their values, we get \(proj_v u= -0.8 \cdot \langle 1,-2\rangle =\langle -0.8,1.6 \rangle\)

Step 3: Find Orthogonal Vector

We know that u can be represented as the sum of its projection onto v and the orthogonal vector. We calculate this orthogonal vector using the formula \(u - proj_v u\). Substituting u and proj_v u by their values, we get the orthogonal vector as \(\langle 4,2\rangle - \langle -0.8, 1.6 \rangle =\langle 4.8, 0.4\rangle\)

Step 4: Represent u as sum of vectors

Finally represent vector u as the sum of these two vectors we calculated: \(u = |\langle -0.8, 1.6\rangle + \langle 4.8, 0.4\rangle|\) which simplifies to \(u = |\langle 4,2 \rangle|\)

Key Concepts

Scalar Projection

The scalar projection is a fundamental concept in vector mathematics. It describes how much of one vector goes in the same direction as another vector. In simpler terms, it's similar to asking, "how much does this vector 'cover' the other vector?" This is a crucial step when trying to deconstruct vector relationships.The scalar projection can be calculated using the formula:

• Take the dot product of the two vectors, \( \mathbf{u} \cdot \mathbf{v} \), which combines them into a single number.
• Divide by the magnitude squared of the vector you're projecting onto, \( ||\mathbf{v}||^2 \). This helps in normalizing the relationship based on the size of the second vector.

Therefore, the scalar projection formula is given by \( c = \frac{\mathbf{u} \cdot \mathbf{v}}{||\mathbf{v}||^2} \).Understanding scalar projection helps us decide the direction and scale of the eventual vector projection. By focusing on the scalar projection, we figure out the alignment and relative positioning of the vectors involved.

Orthogonal Vectors

Orthogonal vectors are vectors that stand at right angles (90 degrees) to each other. This is similar to how the x and y axes intersect in a graph. When working with vectors, finding a pair of orthogonal vectors can help simplify complex geometric problems. A unique property of orthogonal vectors is that their dot product equals zero. This zero dot product indicates that there's no overlap in direction between the vectors. Thus, one vector contributes nothing to the direction of the other. In vector decomposition, one vector is often broken down into two components: a parallel one (or projection) and an orthogonal one. The orthogonal component is significant because it reveals the unique attributes of the vector beyond its projection. Calculating an orthogonal vector involves subtracting the vector projection from the original vector:

• Determine the vector projection first.
• Subtract this projection from the original vector to get the orthogonal vector.

Recognizing orthogonal vectors is beneficial for various applications such as in graphics, physics, and engineering where mutual independence of directions is analyzed.

Vector Operations

Vector operations form the backbone of vector mathematics, enabling us to manipulate and understand vector relationships effectively. Common operations include addition, subtraction, scalar multiplication, and dot product evaluation. Addition and Subtraction:

• Vectors are added or subtracted component-wise, which means we consider each dimension individually.
• This computation helps us move through vector space logically, whether demonstrating overlap or variance in direction and magnitude.

Scalar Multiplication:

• This operation involves multiplying a vector by a scalar, which effectively stretches or shrinks the vector, maintaining the direction but altering its magnitude.

Dot Product:

• Yielding a scalar, the dot product compares how much one vector extends in the direction of another. If vectors are orthogonal, the dot product will be zero, which signifies no directional overlap.

Performing these operations correctly is crucial for various vector applications, from simple projections to complex physical simulations in scientific computing.