Question

Prove (for $E^2$ and $E^3$ ) that $$\overline{x}\cdot \overline{y}=|\overline{x}||\overline{y}|\cos \alpha$$ where $\alpha$ is the angle between the vectors $\overrightarrow{0x}$ and $\overrightarrow{0y}$; we denote $\alpha$ by $⟨\overline{x},\overline{y}⟩.$

Short answer

The dot product formula is verified using cosine definition for both $E^2$ and $E^3$.

Step-by-step solution

Understand the Dot Product

The dot product $\overline{x}\cdot \overline{y}$ of two vectors $\overline{x}$ and $\overline{y}$ in $E^n$ is defined as the sum of the products of their corresponding components if $\overline{x}=(x_1,x_2,...,x_n)$ and $\overline{y}=(y_1,y_2,...,y_n)$. In 2D and 3D, the dot product is $x_1{y}_1+x_2{y}_2$ and $x_1{y}_1+x_2{y}_2+x_3{y}_3$ respectively.

Calculate the Magnitude of Vectors

The magnitude $|\overline{x}|$ of a vector $\overline{x}=(x_1,x_2)$ in $E^2$ is $\sqrt{x_1^2+x_2^2}$ and for $\overline{x}=(x_1,x_2,x_3)$ in $E^3$ it is $\sqrt{x_1^2+x_2^2+x_3^2}$. Similarly, compute $|\overline{y}|$ for vector $\overline{y}$.

Relate Dot Product and Cosine of Angle

By definition, the dot product $\overline{x}\cdot \overline{y}$ can also be expressed in terms of magnitudes and the cosine of the angle $\alpha$ between them: $\overline{x}\cdot \overline{y}=|\overline{x}||\overline{y}|\cos \alpha$. This equation relates the dot product to the geometric interpretation involving angle.

Verify in 2D ($E^2$)

In $E^2$, for $\overline{x}=(x_1,x_2)$ and $\overline{y}=(y_1,y_2)$, the angle cosine can be found from the equation: $$\cos \alpha =\frac{x_1{y}_1+x_2{y}_2}{\sqrt{x_1^2+x_2^2}\cdot \sqrt{y_1^2+y_2^2}}.$$ Substitute the dot product and magnitudes into this equation; this verifies $\overline{x}\cdot \overline{y}=|\overline{x}||\overline{y}|\cos \alpha$.

Verify in 3D ($E^3$)

For vectors $\overline{x}=(x_1,x_2,x_3)$ and $\overline{y}=(y_1,y_2,y_3)$, compute from the formula: $$\cos \alpha =\frac{x_1{y}_1+x_2{y}_2+x_3{y}_3}{\sqrt{x_1^2+x_2^2+x_3^2}\cdot \sqrt{y_1^2+y_2^2+y_3^2}}.$$ When substituted, it will show that $\overline{x}\cdot \overline{y}=|\overline{x}||\overline{y}|\cos \alpha$, thus proving it for $E^3$.

Key concepts

Vector Magnitude

The vector magnitude, sometimes known as the length or norm of a vector, is a measure of the vector's size. For a two-dimensional vector $\overline{x}=(x_1,x_2)$, its magnitude is calculated using the formula $|\overline{x}|=\sqrt{x_1^2+x_2^2}$. This formula derives from the Pythagorean theorem, treating the vector as the hypotenuse of a right triangle formed by its components. In three dimensions, for a vector $\overline{x}=(x_1,x_2,x_3)$, the magnitude extends to include the third component, computed as $|\overline{x}|=\sqrt{x_1^2+x_2^2+x_3^2}$.
Understanding the magnitude of a vector is fundamental in physics and geometry as it represents the true distance of the vector from the origin in the Cartesian coordinate system. Magnitudes are always non-negative, and they provide insight into the vector's scale without regard to direction.

Cosine of Angle

The cosine of an angle between two vectors is a crucial part of understanding their spatial relationship. In the context of vectors $\overline{x}$ and $\overline{y}$, the cosine of the angle $\alpha$ between them is defined by the dot product formula: $\cos \alpha =\frac{\overline{x}\cdot \overline{y}}{|\overline{x}||\overline{y}|}$.
Cosine values range from $-1$ to $1$. A cosine value of $1$ indicates that the vectors are pointing in the same direction (parallel), whereas $-1$ shows they are opposite (anti-parallel). A cosine of $0$ reveals the vectors are perpendicular to one another.

• When $\cos \alpha =1$, vector $\overline{x}$ and vector $\overline{y}$ share the same line of action.
• When $\cos \alpha =0$, they establish the maximum orthogonal separation (90 degrees apart).

By linking the cosine of the angle with the dot product, we can easily determine the relative orientation of vectors based solely on their components.

Geometric Interpretation of Vectors

Vectors are not just random sets of numbers; they have a significant geometric interpretation that simplifies understanding complex spatial relationships. In both two-dimensional $(E^2)$ and three-dimensional $(E^3)$ spaces, vectors can represent position, force, velocity, or other directional quantities.
A geometric interpretation often uses the dot product, representing the extent to which one vector "projects" onto another. When vectors are depicted as arrows in space, the dot product relates to how much of one vector goes in the direction of a second vector, which also determines the angle between them.

• For parallel vectors, the projection is maximal.
• For perpendicular vectors, there is no projection.

This geometric insight is crucial, especially in physics and engineering, where understanding vector directions and magnitudes can affect analysis and problem-solving outcomes. Vectors allow us to model real-world problems with precision and predict the results of various phenomena.