Problem 6 - Calculus: Early Transcendentals

Question

Exercise 12.3.6 Find the cosine of the angle between \langle 1,2,3\rangle and \langle 1,1,1\rangle\(;\) use a calculator if necessary to find the angle.

Step-by-Step Solution

Answer

\(\cos(\theta) \approx 0.928\)

Step 1: Understand the formula for cosine between two vectors

The cosine of the angle \( \theta \) between two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is given by the formula: \[\cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \|\mathbf{b}\|}\] where \( \mathbf{a} \cdot \mathbf{b} \) is the dot product of the vectors and \( \|\mathbf{a}\| \) and \( \|\mathbf{b}\| \) are the magnitudes of \( \mathbf{a} \) and \( \mathbf{b} \) respectively.

Step 2: Compute the dot product

The dot product \( \mathbf{a} \cdot \mathbf{b} \) of vectors \( \mathbf{a} = \langle 1, 2, 3 \rangle \) and \( \mathbf{b} = \langle 1, 1, 1 \rangle \) is calculated as follows: \[1 \times 1 + 2 \times 1 + 3 \times 1 = 1 + 2 + 3 = 6\]

Step 3: Calculate magnitude of vector \( \mathbf{a} \)

The magnitude \( \|\mathbf{a}\| \) of vector \( \mathbf{a} = \langle 1, 2, 3 \rangle \) is determined by: \[\|\mathbf{a}\| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14}\]

Step 4: Calculate magnitude of vector \( \mathbf{b} \)

The magnitude \( \|\mathbf{b}\| \) of vector \( \mathbf{b} = \langle 1, 1, 1 \rangle \) is determined by: \[\|\mathbf{b}\| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{1 + 1 + 1} = \sqrt{3}\]

Step 5: Apply the cosine formula

Substitute the dot product and magnitudes into the cosine formula: \[\cos(\theta) = \frac{6}{\sqrt{14} \times \sqrt{3}} = \frac{6}{\sqrt{42}} = \frac{6}{\sqrt{42}} \approx 0.928\] using a calculator.

Key Concepts

Dot Product

The dot product is a fundamental concept in vector mathematics. It provides a way to multiply two vectors to obtain a scalar (a single number). When trying to understand how it works, think of it as a method to determine how much one vector "points" in the direction of another.

• To compute the dot product, you multiply the corresponding components of two vectors and then sum the results.
• For example, for the vectors \( \mathbf{a} = \langle 1, 2, 3 \rangle \) and \( \mathbf{b} = \langle 1, 1, 1 \rangle \), the calculation is: \( 1\times1 + 2\times1 + 3\times1 = 6 \).

The result, in this case, is 6, which tells us the degree of alignment between the two vectors. When the dot product is positive, the vectors point in somewhat the same direction. A dot product of zero indicates that the vectors are orthogonal (right angles to each other), showing no direct alignment. The negative dot product would reflect opposite directions.

Magnitude of a Vector

The magnitude of a vector is like the "length" or "size" of the vector. This measurement is crucial because it quantifies how far the vector extends in space, regardless of direction.

• To find the magnitude of a vector, you take the square root of the sum of the squares of its components.
• For the vector \( \mathbf{a} = \langle 1, 2, 3 \rangle \), the magnitude is \( \|\mathbf{a}\| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{14} \).
• Similarly, for \( \mathbf{b} = \langle 1, 1, 1 \rangle \), the magnitude is \( \|\mathbf{b}\| = \sqrt{3} \).

This understanding is significant in vector mathematics as it allows us to scale vectors and understand their geometric properties better. Knowing the magnitude is also essential when calculating other attributes like the cosine of the angle between vectors.

Cosine of an Angle

The cosine of the angle between two vectors helps in determining how aligned or opposing the vectors are in terms of direction. It's a measure that ranges from -1 to 1, where 1 implies identical direction, -1 means opposite directions, and 0 indicates perpendicularity.

• The formula to find the cosine of the angle \( \theta \) between two vectors involves their dot product and magnitudes: \( \cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \|\mathbf{b}\|} \).
• In our example, with \( \mathbf{a} = \langle 1, 2, 3 \rangle \) and \( \mathbf{b} = \langle 1, 1, 1 \rangle \), we've calculated \( \mathbf{a} \cdot \mathbf{b} = 6 \), \( \|\mathbf{a}\| = \sqrt{14} \), and \( \|\mathbf{b}\| = \sqrt{3} \).
• Applying these numbers to the formula gives \( \cos(\theta) = \frac{6}{\sqrt{42}} \).

Using a calculator, you can find that \( \cos(\theta) \approx 0.928 \). This value indicates that the vectors are pointing in a similar direction, as the cosine is closer to 1. Understanding this concept enables us to analyze and interpret the directional relationship between different vectors effectively.