Problem 10 - Essential Calculus Early Transcendentals

Question

2-10 Find \(\mathbf{a} \cdot \mathbf{b}\) $$|\mathbf{a}|=3, \quad|\mathbf{b}|=\sqrt{6}, \quad$$ the angle between a and \(\mathbf{b}\) is \(45^{\circ}\)

Step-by-Step Solution

Answer

\(\mathbf{a} \cdot \mathbf{b} = 3 \sqrt{3}\)

Step 1: Understanding the Dot Product Formula

The dot product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is defined as \( \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta \), where \( \theta \) is the angle between the vectors. We need to apply this formula with the given magnitudes and angle.

Step 2: Substituting Known Values

Use the magnitudes \( |\mathbf{a}| = 3 \) and \( |\mathbf{b}| = \sqrt{6} \), and the angle \( \theta = 45^{\circ} \). Substitute these values into the dot product formula:\[\mathbf{a} \cdot \mathbf{b} = 3 \times \sqrt{6} \times \cos(45^{\circ})\]

Step 3: Calculating Cosine of the Angle

The cosine of \( 45^{\circ} \) is \( \frac{\sqrt{2}}{2} \). Substitute this value into the equation:\[\mathbf{a} \cdot \mathbf{b} = 3 \times \sqrt{6} \times \frac{\sqrt{2}}{2}\]

Step 4: Performing the Multiplication

Calculate the product of the terms:\[\mathbf{a} \cdot \mathbf{b} = \left( 3 \times \frac{\sqrt{2}}{2} \right) \times \sqrt{6} = \frac{3 \sqrt{12}}{2}\]

Step 5: Simplifying the Expression

Simplify \( \sqrt{12} \) which is equal to \( \sqrt{4 \times 3} = 2\sqrt{3} \). Substitute back and simplify:\[\mathbf{a} \cdot \mathbf{b} = \frac{3 \times 2 \sqrt{3}}{2} = 3 \sqrt{3}\]

Key Concepts

Vector Magnitude

Vector magnitude is a measure of a vector's length or size. It tells us how "big" the vector is without considering its direction. We often denote the magnitude of a vector \( \mathbf{a} \) as \( |\mathbf{a}| \).

To find the magnitude of a vector, you can use the formula:

• For a vector in 2D, \( \mathbf{a} = (x, y) \), its magnitude is \( |\mathbf{a}| = \sqrt{x^2 + y^2} \).
• In 3D, \( \mathbf{a} = (x, y, z) \), it becomes \( |\mathbf{a}| = \sqrt{x^2 + y^2 + z^2} \).

In our example, the magnitude of \( \mathbf{a} \) is given as 3 and the magnitude of \( \mathbf{b} \) is \( \sqrt{6} \). These values help us compute their dot product, which involves multiplying their magnitudes together along with the cosine of the angle between them. Understanding magnitude is crucial, as it forms the basis of other vector calculations.

Angle Between Vectors

The angle between vectors is a key aspect of understanding how vectors relate to each other geometrically. It reveals not only the direction but how vectors "align." The smaller the angle, the more aligned or parallel they are, whereas larger angles indicate vectors pointing away from each other.

We use the dot product formula to find this angle, given by the relationship:\[ \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|mathbf{a}| |mathbf{b}|} \]

In this exercise, we know \( \theta = 45^{\circ} \), which is a commonly known angle in trigonometry.

• An angle of \( 45^{\circ} \) often corresponds to common trigonometric values, making calculations like the cosine straightforward.

Relevance in Calculations

The angle plays a direct role in determining the value of the dot product, as it adjusts the contribution of the magnitudes based on directional alignment. So, being clear about how to use angle information is key for mastering vector operations.

Cosine Function

The cosine function is an essential trigonometric function that facilitates the calculation of angles. It connects the adjacent side of a right triangle to its hypotenuse. Particularly important in vector mathematics, it helps assess how aligned two vectors are.

The cosine of an angle \( \theta \) is expressed as:

• \( \cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}} \) in a right triangle.
• Common angles have known cosine values, such as \( \cos(0^{\circ}) = 1 \), \( \cos(45^{\circ}) = \frac{\sqrt{2}}{2} \), and \( \cos(90^{\circ}) = 0 \).

Application in Vectors

When using the dot product formula, the cosine of the angle between vectors helps determine how much one vector "projects" onto another. If two vectors make an angle of \( 45^{\circ} \), their cosine helps illustrate that they're neither completely aligned nor completely perpendicular.

Understanding the cosine function is vital for calculating and using angles between vectors, an essential part of many physical and geometrical interpretations.