Problem 10 - University Calculus: Early Transcendentals

Question

Find the angles between the vectors in Exercises \(9-12\) to the nearest hundredth of a radian. $$\mathbf{u}=2 \mathbf{i}-2 \mathbf{j}+\mathbf{k}, \quad \mathbf{v}=3 \mathbf{i}+4 \mathbf{k}$$

Step-by-Step Solution

Answer

The angle between the vectors is approximately 0.84 radians.

Step 1: Determine the dot product

Calculate the dot product of the given vectors \(\mathbf{u}\) and \(\mathbf{v}\). The dot product is given by the formula \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 \), where \(\mathbf{u} = (u_1, u_2, u_3)\) and \(\mathbf{v} = (v_1, v_2, v_3)\). Substitute the given components: \(2\times3 + (-2)\times0 + 1\times4 = 6 + 0 + 4 = 10\). Thus, the dot product is 10.

Step 2: Calculate the magnitudes

Find the magnitude of each vector using the formula \( \|\mathbf{u}\| = \sqrt{u_1^2 + u_2^2 + u_3^2} \) and \( \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2} \). For \(\mathbf{u}\), \(\|\mathbf{u}\| = \sqrt{2^2 + (-2)^2 + 1^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3\). For \(\mathbf{v}\), \(\|\mathbf{v}\| = \sqrt{3^2 + 0^2 + 4^2} = \sqrt{9 + 0 + 16} = \sqrt{25} = 5\).

Step 3: Use the dot product formula to find the angle

The angle \( \theta \) between two vectors can be found using the formula \( \cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|} \). Substitute the values: \( \cos \theta = \frac{10}{3 \times 5} = \frac{10}{15} = \frac{2}{3} \).

Step 4: Calculate the angle

Solve for \( \theta \) by taking the arccosine of \(\frac{2}{3}\): \( \theta = \cos^{-1}\left(\frac{2}{3}\right) \). Using a calculator, \( \theta \approx 0.8411 \) radians.

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in vector algebra. It is performed by multiplying corresponding components of two vectors and then adding up all the results. The formula for the dot product of two vectors, \( \mathbf{u} = (u_1, u_2, u_3) \) and \( \mathbf{v} = (v_1, v_2, v_3) \), is:

• \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 \)

This results in a single scalar value rather than a vector.
The dot product can illustrate whether vectors point in a similar direction. A positive dot product suggests that the vectors point more in the same direction, while a negative result indicates the opposite. If the dot product is zero, the vectors are perpendicular to each other.

Vector Magnitude

Understanding the magnitude of a vector is crucial because it expresses the "length" of the vector, regardless of its direction.

• The magnitude of a vector \( \mathbf{u} = (u_1, u_2, u_3) \) is calculated as \( \| \mathbf{u} \| = \sqrt{u_1^2 + u_2^2 + u_3^2} \).
• Similarly, for vector \( \mathbf{v} = (v_1, v_2, v_3) \), it is \( \| \mathbf{v} \| = \sqrt{v_1^2 + v_2^2 + v_3^2} \).

The magnitude helps to understand how "strong" or "intense" a vector is.
In calculations of vector angles, knowing the magnitude of each vector is essential for determining the cosine of the angle they form.

Cosine of Angle

The cosine of the angle between two vectors can be discovered using the dot product and the magnitudes of the vectors.

• The formula is \( \cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{\| \mathbf{u} \| \| \mathbf{v} \|} \).

When you divide the dot product of two vectors by the product of their magnitudes, you get the cosine of the angle \( \theta \).
The cosine value helps to determine how much the two vectors point in the same or opposite directions.
Values closer to 1 mean the vectors are almost parallel, while values closer to -1 imply they are opposite.
A value around 0 describes vectors that are nearly perpendicular.

Arccosine Function

The arccosine function is a trigonometric function, often written as \( \cos^{-1} \).
It is used to find the angle whose cosine is a given value.

• For instance, if \( \cos \theta = \frac{2}{3} \), the corresponding angle \( \theta \) is calculated as \( \theta = \cos^{-1}\left(\frac{2}{3}\right) \).

The result derived from the arccosine is usually in the range of 0 to \( \pi \) radians, which corresponds to angles from 0 to 180 degrees.
This function is crucial for solving problems that involve finding the angle between vectors, giving you precise angle measures.