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The dot product, also known as the scalar product, is a fundamental operation in vector algebra. This operation combines two vectors to produce a single scalar value. The formula for the dot product of two vectors \( \mathbf{u} = (u_1, u_2) \) and \( \mathbf{v} = (v_1, v_2) \) is given by:\[\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 \]

• This calculation involves multiplying each corresponding component of the vectors and then adding the results.
• The dot product is a versatile tool as it can reveal whether two vectors are perpendicular. If the dot product of two vectors is zero, the vectors are orthogonal.

In the case of vectors \( \mathbf{u} = 2 \mathbf{i} - 3 \mathbf{j} \) and \( \mathbf{v} = \mathbf{i} - 2 \mathbf{j} \), the calculation of their dot product is as follows:- Multiply the \( \mathbf{i} \) components: \( 2 \times 1 = 2 \)- Multiply the \( \mathbf{j} \) components: \( -3 \times -2 = 6 \)- Add these products: \( 2 + 6 = 8 \)The scalar result here, 8, is key to understanding the relationship between the vectors in later steps.

The magnitude of a vector, also known as the vector's length, measures how long the vector is. It is calculated using the Pythagorean theorem for vectors in two dimensions. For a vector \( \mathbf{u} = (u_1, u_2) \), the magnitude \(||\mathbf{u}||\) is determined by:\[||\mathbf{u}|| = \sqrt{u_1^2 + u_2^2} \]

• This formula involves squaring each component of the vector, adding them together, and taking the square root of the sum.
• The magnitude is always a non-negative value, representing the distance of the vector's tip from the origin when plotted on a graph.

For the vectors given, \( \mathbf{u} = 2 \mathbf{i} - 3 \mathbf{j} \) and \( \mathbf{v} = \mathbf{i} - 2 \mathbf{j} \):- The magnitude of \( \mathbf{u} \) is \( \sqrt{2^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \).- The magnitude of \( \mathbf{v} \) is \( \sqrt{1^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \).These magnitudes are crucial when calculating the angle between the vectors.

The inverse cosine function, often written as \( \cos^{-1} \) or \( \arccos \), is used to find the angle whose cosine is a given number. This mathematical function is particularly useful when wanting to find the angle between two vectors.

• To find the angle \( \theta \) between two vectors, use the equation: \[\cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{||\mathbf{u}|| ||\mathbf{v}||} \]
• The result of the dot product divided by the product of the magnitudes gives the cosine of the angle \( \theta \).
• To extract the actual angle, apply the inverse cosine.

Substituting our calculated values for \( \mathbf{u} \cdot \mathbf{v} = 8 \), \(||\mathbf{u}|| = \sqrt{13} \), and \(||\mathbf{v}|| = \sqrt{5} \):\[ \cos(\theta) = \frac{8}{\sqrt{13} \cdot \sqrt{5}} \]Finally, to find \( \theta \), compute \( \theta = \cos^{-1}\left( \frac{8}{\sqrt{13} \cdot \sqrt{5}} \right) \).Ensure your calculator is set correctly to either radians or degrees, depending on the required units for the answer. This final step transforms the dot product and magnitudes into an understandable angle measurement.