Problem 4 - Active Calculus Multivariable

Question

Find \(\mathbf{a} \cdot \mathbf{b}\) if \(|\mathbf{a}|=7,|\mathbf{b}|=7,\) and the angle between \(\mathbf{a}\) and \(\mathbf{b}\) is \(-\frac{\pi}{10}\) radians. \(\mathbf{a} \cdot \mathbf{b}=\) _________

Step-by-Step Solution

Answer

\(\mathbf{a} \cdot \mathbf{b} \approx 46.6039\)

Step 1: Recall the formula for the dot product in terms of magnitudes and angle

The dot product of two vectors can be calculated using their magnitudes and the angle between them using the following formula: \(\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| \cdot |\mathbf{b}| \cdot \cos(\theta)\).

Step 2: Substitute the given magnitudes and angle into the formula

We are given the magnitudes: \(|\mathbf{a}| = 7\), \(|\mathbf{b}| = 7\), and the angle between them: \(\theta = -\frac{\pi}{10}\). We substitute these values into the formula: \(\mathbf{a} \cdot \mathbf{b} = 7 \cdot 7 \cdot \cos\left(-\frac{\pi}{10}\right)\).

Step 3: Evaluate the cosine of the angle and simplify the expression

Calculate the cosine of the angle and simplify the expression: \(\mathbf{a} \cdot \mathbf{b} = 7 \cdot 7 \cdot \cos\left(-\frac{\pi}{10}\right) = 49 \cdot \cos\left(-\frac{\pi}{10}\right)\). Since the cosine function is even, we have: \(\cos\left(-\frac{\pi}{10}\right) = \cos\left(\frac{\pi}{10}\right)\). Hence, \(\mathbf{a} \cdot \mathbf{b} = 49 \cdot \cos\left(\frac{\pi}{10}\right)\).

Step 4: Find the numerical value of the dot product

We can now calculate the numerical value of the dot product: \(\mathbf{a} \cdot \mathbf{b} = 49 \cdot \cos\left(\frac{\pi}{10}\right) \approx 49 \cdot 0.9511 = 46.6039\). Therefore, \(\mathbf{a} \cdot \mathbf{b} \approx 46.6039\).

Key Concepts

Vector Magnitudes

Vectors represent quantities that have both a magnitude and a direction. The magnitude of a vector can be thought of as its length. It is a measure of how long the vector is. Often denoted by the symbol \( |\mathbf{a}| \), vector magnitudes are crucial when calculating the dot product of two vectors.

• Magnitude Calculation: For a vector \(\mathbf{a} = (x, y, z)\), the magnitude is calculated using the formula: \( |\mathbf{a}| = \sqrt{x^2 + y^2 + z^2} \).
• Importance: Knowing the magnitude helps determine how "big" or "small" a vector is, without considering its direction.

In this exercise, we know both vectors \(\mathbf{a}\) and \(\mathbf{b}\) have magnitudes of 7. This information is used to find their dot product.

Angle Between Vectors

The angle between two vectors is essential for understanding their orientation relative to each other. This concept is closely related to the dot product. The angle, often represented by \(\theta\), determines how much one vector has to rotate to align with another vector.

• Range: The angle \(\theta\) can range from \(0\) to \(\pi\) radians.
• Significance: This angle helps define the cosine value needed for the dot product calculation.

In our problem, the given angle is \(-\frac{\pi}{10}\) radians, which translates to a positive angle \(\frac{\pi}{10}\) for the cosine calculation due to the even nature of the cosine function.

Trigonometric Functions

Trigonometric functions are used to relate the angles and lengths of triangles. In vector mathematics, they help establish relationships between the vectors via functions like sine, cosine, and tangent.

• Cosine Function: Particularly useful in the context of dot products, as it provides the ratio of the adjacent side to the hypotenuse in a right triangle encompassing angle \(\theta\).
• Applications: These functions enable calculations involving angles and can derive meaningful geometric and algebraic information about vectors.

In this exercise, the trigonometric function we use is the cosine, which gives us the information needed to perform the dot product calculation efficiently.

Cosine Function

The cosine function is a key mathematical function used to find the dot product between two vectors. It calculates the cosine of an angle, which determines how two vectors align with each other.

• Definition: For any given angle \(\theta\), \(\cos(\theta)\) is the x-coordinate of the point on the unit circle at that angle.
• Even Function: Cosine is an even function, meaning \(\cos(\theta) = \cos(-\theta)\). This property simplifies computations, as seen when the angle is \(-\frac{\pi}{10}\).
• Importance in Dot Product: Through \(\cos(\theta)\), the dot product becomes a measure of not just how aligned vectors are, but how much they "project" onto each other.

Knowing that \(\cos(-\frac{\pi}{10}) = \cos(\frac{\pi}{10})\) allowed us to convert our problem to a standard angle and make the calculation straightforward in finding the dot product.