Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the cosine of the angle between \(\mathbf{a}\) and \(\mathbf{b}\) and make a sketch. (a) \(\mathbf{a}=\langle 1,-3\rangle, \mathbf{b}=\langle-1,2\rangle\) (b) \(\mathbf{a}=\langle-1,-2\rangle, \mathbf{b}=\langle 6,0\rangle\) (c) \(\mathbf{a}=\langle 2,-1\rangle, \mathbf{b}=\langle-2,-4\rangle\) (d) \(\mathbf{a}=\langle 4,-7\rangle, \mathbf{b}=\langle-8,10\rangle\)

Step-by-Step Solution

Answer

Calculate the dot product and magnitudes, then use the formula \( \cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{\| \mathbf{a} \| \| \mathbf{b} \|} \) to find the cosine for each vector pair.

Step 1: Understand the formula for cosine of the angle between two vectors

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

Step 2: Calculate the dot product of vectors

The dot product of two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \) is calculated as:\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \]Apply this to each part (a)-(d) of the question.

Step 3: Calculate the magnitudes of the vectors

The magnitude of a vector \( \mathbf{a} = \langle a_1, a_2 \rangle \) is:\[ \| \mathbf{a} \| = \sqrt{a_1^2 + a_2^2} \]Calculate the magnitudes of \( \mathbf{a} \) and \( \mathbf{b} \) for each part (a)-(d).

Step 4: Plug values into the cosine formula

For each part (a)-(d), substitute the calculated dot product and magnitudes into the cosine formula:\[ \cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{\| \mathbf{a} \| \times \| \mathbf{b} \|} \]

Step 5: Simplify each expression to find cos(θ)

Simplify the expression from Step 4 for each part (a)-(d) to get the cosine of the angle between the vectors.

Step 6: Sketch the vectors

Draw a simple sketch of vectors \( \mathbf{a} \) and \( \mathbf{b} \) for each part (a)-(d) on a coordinate plane, ensuring to include their approximate directions and relative angles.

Key Concepts

Dot Product

The dot product is a fundamental concept in vector algebra and is used to find relationships between vectors. For two vectors, \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \), the dot product \( \mathbf{a} \cdot \mathbf{b} \) is calculated with the formula:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \)

This formula essentially multiplies the corresponding components of the two vectors and sums up those products.
A higher positive value of the dot product indicates that the vectors point more in the same direction. Conversely, if the dot product is zero, it means that the vectors are perpendicular to each other.

Understanding the dot product allows you to determine various properties about the angle between two vectors, which is essential in topics such as physics and engineering where vector calculations are common.
It's particularly crucial when calculating the cosine of the angle between the vectors, as seen in exercises such as finding \( \cos(\theta) \).

Vector Magnitude

The magnitude of a vector is essentially its length in the coordinate system. For a vector \( \mathbf{a} = \langle a_1, a_2 \rangle \), the magnitude \( \|\mathbf{a}\| \) is calculated using the formula:

• \( \|\mathbf{a}\| = \sqrt{a_1^2 + a_2^2} \)

This formula ensures you are taking the "square root of the sum of squares" of its components.
It’s equivalent to finding the hypotenuse in a right-angled triangle represented by the vector's components.

Magnitude is vital when determining the distance between points in space and plays a key role in normalizing vectors.
Normalization is the process of scaling a vector to have a magnitude of one, often needed in computer graphics and physics simulations to simply represent directions.

To find the angle between two vectors, knowing their magnitudes allows us to use them in the cosine formula and complete the vector comparison.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves the study of geometric figures through an algebraic lens using a coordinate system. This branch of mathematics allows us to visualise and solve geometric problems using algebra.
Vectors are points that have both a direction and a magnitude in coordinate geometry.

The geometric interpretation of vector operations, such as the dot product or calculating magnitudes, often involves placing these calculations into a coordinate plane.
Here, it becomes possible to draw vectors as arrows, helping to visually determine aspects such as their angle or direction.

For exercises requiring sketches, such as determining the cosine of angles between vectors, coordinate geometry plays a crucial role.
By sketching vectors \( \mathbf{a} \) and \( \mathbf{b} \) on this plane, it becomes easier to comprehend the spatial relationships between them, enhancing understanding over purely numerical computation.
Thus, coordinate geometry serves as the bridge connecting algebraic vector operations to intuitive geometrical visualisations.