Problem 12 - Precalculus Mathematics for Calculus

Question

Find \((a) u \cdot v\) and \((b)\) the angle between \(u\) and \(v\) to the nearest degree. $$\mathbf{u}=\mathbf{i}+\mathbf{j}, \quad \mathbf{v}=\mathbf{i}-\mathbf{j}$$

Step-by-Step Solution

Answer

The dot product is 0, and the angle between the vectors is 90°.

Step 1: Find the Dot Product

The dot product of vectors \( \mathbf{u} = \mathbf{i} + \mathbf{j} \) and \( \mathbf{v} = \mathbf{i} - \mathbf{j} \) is calculated using the formula:\[\mathbf{u} \cdot \mathbf{v} = (i \cdot i) + (j \cdot -j) = 1 \times 1 + 1 \times (-1) = 1 - 1 = 0.\]

Step 2: Find the Magnitudes of Vectors

The magnitude of vector \( \mathbf{u} \) is:\[|\mathbf{u}| = \sqrt{i^2 + j^2} = \sqrt{1^2 + 1^2} = \sqrt{2}.\]The magnitude of vector \( \mathbf{v} \) is:\[|\mathbf{v}| = \sqrt{i^2 + (-j)^2} = \sqrt{1^2 + 1^2} = \sqrt{2}.\]

Step 3: Calculate the Angle Between the Vectors

The cosine of the angle \( \theta \) between the vectors is given by the formula:\[\cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| |\mathbf{v}|}.\]Plugging in the values:\[\cos \theta = \frac{0}{\sqrt{2} \times \sqrt{2}} = 0.\]Thus, \( \theta = \cos^{-1}(0) = 90^{\circ} \).

Key Concepts

Understanding Vectors

Vectors are fundamental in mathematics and physics as they allow us to present quantities that have both magnitude and direction. A vector is usually represented with an arrow; the length of the arrow indicates the magnitude, while the direction of the arrow shows the direction of the vector.
In a coordinate system, a vector can be expressed as a combination of its components along the axes. For example, vector \( \mathbf{u} = \mathbf{i} + \mathbf{j} \) is a simple representation where \( \mathbf{i} \) and \( \mathbf{j} \) are the unit vectors along the x and y axes, respectively.
Vectors can be added and subtracted. They also play a crucial role in calculating quantities such as displacements, velocities, and forces in physics.

Determining the Angle Between Vectors

The angle between two vectors is a measure of how diverged or converged they are in their directions. To find this angle, we use the dot product of the vectors and their magnitudes.
The dot product of two vectors \( \mathbf{u} \) and \( \mathbf{v} \) is calculated as the sum of the product of their corresponding components. It is denoted as \( \mathbf{u} \cdot \mathbf{v} \).
When the dot product is zero, it implies that the vectors are perpendicular to each other. This is because their cosine relationship becomes zero, resulting in a \( 90^{\circ} \) angle.

• Suppose \( \mathbf{u} = \mathbf{i} + \mathbf{j} \) and \( \mathbf{v} = \mathbf{i} - \mathbf{j} \). Their dot product is zero as seen in the provided example. Therefore, the angle between them is \( 90^{\circ} \).

Understanding this concept helps in several calculations involving physics, such as determining orthogonal forces.

Measuring the Magnitude of a Vector

The magnitude of a vector is essentially its length. It is calculated using the Pythagorean theorem when the vector is in a plane. If a vector has components \( x \) and \( y \), its magnitude \(|\mathbf{u}|\) is computed by \( \sqrt{x^2 + y^2} \).
Magnitude provides a scalar value showing the size of the vector without considering the direction.

• For example, the magnitude of \( \mathbf{u} = \mathbf{i} + \mathbf{j} \), using the formula, results in \( \sqrt{1^2 + 1^2} = \sqrt{2} \).

This calculation is essential in physics and engineering, where magnitude is needed to determine things like speed and force, independent of direction.