91Ó°ÊÓ

The dot product, also known as the scalar product, is a fundamental operation in vector algebra. It's essential in determining the relationship between two vectors.
To compute the dot product of two vectors, you multiply their corresponding components and sum the results. For vectors \( \overrightarrow{a} = (a_1, a_2) \) and \( \overrightarrow{b} = (b_1, b_2) \), the dot product is calculated as:
\[ \overrightarrow{a} \cdot \overrightarrow{b} = a_1b_1 + a_2b_2 \]
In the exercise, \( \overrightarrow{OP} \cdot \overrightarrow{OQ} = 4 \cdot 5 + 5 \cdot (-7) = 20 - 35 = -15 \). The dot product is a scalar quantity, and here, it helped in determining the angle between the vectors in the later steps.

Determining the angle between two vectors is a crucial concept in geometry and physics. It tells us how the vectors relate spatially to each other.
The formula to find the angle \( \theta \) between two vectors \( \overrightarrow{a} \) and \( \overrightarrow{b} \) is:
\[ \cos \theta = \frac{\overrightarrow{a} \cdot \overrightarrow{b}}{\|\overrightarrow{a}\| \|\overrightarrow{b}\|} \]
Here, \( \|\overrightarrow{a}\| \) and \( \|\overrightarrow{b}\| \) are the magnitudes of the vectors. Using the inverse cosine function, \( \theta = \cos^{-1}\left(\frac{-15}{\sqrt{41}\cdot\sqrt{74}}\right) \), given in the exercise, we find \( \theta \approx 99.87^\circ \).
This angle tells us that \( \overrightarrow{OP} \) and \( \overrightarrow{OQ} \) form an obtuse angle, indicating they are pointing in significantly different directions.

Unit vectors are vectors with a magnitude of 1. They are mainly used to indicate direction and are essential in expressing any vector in a standardized form.
In a Cartesian coordinate system, the standard unit vectors are \( \hat{i} \) and \( \hat{j} \), which point in the direction of the x-axis and y-axis, respectively.
Any vector can be expressed as a combination of these unit vectors. For example, in the exercise, \( \overrightarrow{OP} = 4\hat{i} + 5\hat{j} \) and \( \overrightarrow{OQ} = 5\hat{i} - 7\hat{j} \).
The use of unit vectors makes it easy to visualize and perform operations such as addition, subtraction, and finding projections in 2D space.

The magnitude of a vector is a measure of its length and is calculated similarly to the hypotenuse of a right-angled triangle. The formula for a vector \( \overrightarrow{a} = (a_1, a_2) \) is:
\[ \|\overrightarrow{a}\| = \sqrt{a_1^2 + a_2^2} \]
This formula derives from the Pythagorean theorem. In the provided exercise, the magnitudes were calculated for vectors \( \overrightarrow{OP} = (4, 5) \) as \( \sqrt{41} \) and \( \overrightarrow{OQ} = (5, -7) \) as \( \sqrt{74} \).
A vector's magnitude is always a non-negative scalar value, representing how far the vector stretches from its point of origin, typically the origin in a coordinate plane.