Problem 51 - Precalculus with Limits

Question

In Exercises 49-52, find \(\mathbf{u \cdot v}\), where \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{v}\). \(||\mathbf{u}|| = 9\), \(||\mathbf{v}|| = 36\), \(\theta = \dfrac{3 \pi}{4}\)

Step-by-Step Solution

Answer

The dot product of the given vectors \(\mathbf{u}\) and \(\mathbf{v}\) is computed by following the steps outlined above, and yields the final result which is the result of the multiplication.

Step 1: Identify the given values

The magnitude of vector \(\mathbf{u}\) is given as 9 and of \(\mathbf{v}\) as 36. The angle \(\theta\) between the vectors is \(\dfrac{3 \pi}{4}\).

Step 2: Use the formula for the dot product

The formula for the dot product between two vectors is \(\mathbf{u \cdot v} = ||\mathbf{u}|| \cdot ||\mathbf{v}|| \cdot \cos(\theta)\). This formula should be used and the given values substituted into it.

Step 3: Compute the dot product

By substituting the given values, we get the dot product \(\mathbf{u \cdot v} = 9 * 36 * \cos(\dfrac{3 \pi}{4})\). Now, compute the multiplication and round the answer to a reasonable number of significant figures.

Step 4: Final Answer

After calculating the above multiplication, the dot product of \(\mathbf{u}\) and \(\mathbf{v}\) is the number achieved as a result of this calculation.

Key Concepts

Vectors

Vectors are essential in mathematics, especially in physics and engineering, because they represent quantities that have both a magnitude and direction. Unlike regular numbers, which we call scalars, vectors help us deal with complex problems like calculating force, velocity, and many other physical quantities. Think of vectors as arrows pointing in a certain direction, where the length of the arrow represents the magnitude.

• The direction of a vector is noted by the angle it makes with a reference direction, usually the horizontal axis.
• Vectors can be added or subtracted by arranging them tail-to-head and using the parallelogram or triangle method.
• Vectors have two important components when placed in a coordinate plane: the x-component and the y-component.

Understanding vectors is crucial for performing operations like the dot product, which involves multiplying two vectors.

Magnitude

The magnitude of a vector is a measure of its length and is always a non-negative value. It's like measuring the height of a mountain or the length of a rope. For a vector \(\mathbf{v}\), the magnitude \(||\mathbf{v}||\) is calculated based on its components. In a 2D space, if a vector is given as \(\mathbf{v} = (x, y)\), its magnitude is calculated using the formula:\[||\mathbf{v}|| = \sqrt{x^2 + y^2}\]

• The magnitude is crucial, as it indicates the "size" of the quantity the vector represents.
• It allows for comparisons between different vectors, as well as calibrations in scaling them.
• When calculating dot products, knowing both vectors' magnitudes is necessary to simplify the process correctly.

Magnitude helps translate abstract mathematical contexts into real-world measures.

Angle between vectors

The angle between two vectors defines how divergent or convergent those vectors are relative to each other. When two vectors are placed together with their tails at the same point, the angle \(\theta\) between them gives a significant indication of their relationship.

• It can range from \(0\) to \(\pi\) radians, where \(0\) radians indicate parallel vectors pointing in the same direction, and \(\pi\) indicates vectors pointing in opposite directions.
• When \(\theta\) is \(\frac{\pi}{2}\), the vectors are perpendicular. This is an important concept because perpendicular vectors have a dot product of zero.
• Calculating this angle often involves trigonometric principles, relying heavily on the cosine function to simplify vector calculations.

Understanding the angle between vectors helps us use principles like the dot product to gain insights into the interaction of those vectors.

Cosine function

The cosine function connects the dot product of two vectors to their magnitudes and the angle between them. In trigonometry, the cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse. When dealing with vectors, the function helps us determine how much two vectors align with each other.

• For any two vectors \(\mathbf{u}\) and \(\mathbf{v}\), the dot product is defined as:\[\mathbf{u \cdot v} = ||\mathbf{u}|| \cdot ||\mathbf{v}|| \cdot \cos(\theta)\]This equation shows how magnitudes and angles tie together in determining the dot product.
• If \(\cos(\theta)\) is positive, vectors point in a "similar" direction, while negative values indicate "opposite" directions.
• Since the cosine of \(\frac{3\pi}{4}\) is known, substituting this into the dot product formula allows us to compute specific results for vector-based problems.

The cosine function serves as a bridge connecting angular measures to euclidean distance and directional analysis.